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Multivariate extremal dependence and risk measures Hua, Lei 2012

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Multivariate Extremal Dependence and Risk Measures  by Lei Hua B.Econ., University of Science and Technology of China, 2002 M.Sc. University of Calgary, 2008  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  Doctor of Philosophy in THE FACULTY OF GRADUATE STUDIES (Statistics)  The University of British Columbia (Vancouver) June 2012 c Lei Hua, 2012  Abstract Overlooking non-Gaussian and tail dependence phenomena has emerged as an important reason of underestimating aggregate financial or insurance risks. For modeling the dependence structures between non-Gaussian random variables, the concept of copula plays an important role and provides practitioners with promising quantitative tools. In order to study copula families that have different tail patterns and tail asymmetry than multivariate Gaussian and t copulas, we introduce the concepts of tail order and tail order functions. These provide a unified way to study three types of dependence in the tails: tail dependence, intermediate tail dependence and tail orthant independence. Some fundamental properties of tail order and tail order functions are obtained. For multivariate Archimedean copulas, we relate the tail heaviness of a positive random variable to the tail behavior of the Archimedean copula constructed by the Laplace transform of the random variable. Quantitative risk measurements pay more attention on large losses. A good statistical approach for the whole data does not guarantee a good way for risk assessments. We use tail comonotonicity as a conservative dependence structure for modeling multivariate dependent losses. By this way, we do not lose too much accuracy but gain reasonable conservative risk measures, especially when we consider high-risk scenarios. We have conducted a thorough investigation on the properties and constructions of tail comonotonicity, and found interesting properties such as asymptotic additivity properties of risk measures. Sufficient conditions have also been obtained to justify the conservativity of tail comonotonicity. For large losses, tail behavior of loss distributions is more critical than the whole distributions. Asymptotic study assuming that each marginal risk goes to infinity is more mathematically tractable. However, the asymptotic study that leads ii  to a first order approximation is only a crude way and may not be sufficient. To this end, we study the second order conditions for risk measures of sub-extremal multiple risks. Some relationships between Value at Risk and Conditional Tail Expectation have been obtained under the condition of Second Order Regular Variation. We also find that the second order parameter determines whether a higher order approximation is necessary.  iii  Preface This thesis has been written up under the supervision of my advisor Professor Harry Joe, and it is based on four papers coauthored with Dr. Harry Joe. Two of them have been published, one is to be published and the other is under review. This thesis also contains a small portion of research that has not been published or submitted anywhere by the date of completing the thesis. Chapter 1 was prepared for the purpose of presenting the general shape or taste of the main research within the thesis. More than half of the contents are newly written and the others are scattered in the following Chapters 3 to 6. Chapter 3 is based on a published paper: Hua and Joe (2011a), Tail order and intermediate tail dependence of multivariate copulas. Journal of Multivariate Analysis, 102: 1454–1471. This chapter covers the topics in the paper but not exclusively. Dr. Joe identified the main research questions and I contributed to the paper mainly through completing the majority of the proofs, writing the first version of the manuscripts and conducting follow-up revisions. The key idea for the proof of the important Lemma 3.4 was contributed by Dr. Joe, and the proof for Proposition 3.5 also benefited from the idea. Chapter 4 is based on a submitted paper: Hua and Joe (2012b), Tail comonotonicity: properties, constructions, and asymptotic additivity of risk measures. The concept of tail comonotonicity was partially motivated by Dr. Joe, especially with his concrete examples. I proposed some main ideas such as the asymptotic additivity of risk measures, completing almost all of the proofs, writing the first version of the manuscripts and taking care of follow-up tasks. Chapter 5 is based on a paper in press: Hua and Joe (2012a), Tail comonotonicity and conservative risk measures. to appear in ASTIN Bulletin, 2012. I iv  conducted all of the research here, including theoretical development, simulation study and data analysis under the supervision of Dr. Joe. I wrote the first version of the manuscripts and conducted follow-up tasks. Chapter 6 is based on a published paper: Hua and Joe (2011b), Second order regular variation and conditional tail expectation of multiple risks. Insurance Math. Econom., 49: 537–546, 2011. I proposed the main ideas and conducted all of the research, including mathematical proofs and simulations under the supervision of Dr. Joe. I wrote up the first version of the manuscripts and dealt with other followup tasks. For all of the above referred research, in addition to the contributions mentioned, Dr. Joe spent a huge amount of time motivating research questions, checking the results and proofs, improving the writing and making all the other contributions to make the research contained in this thesis better.  v  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  x  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xii  Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xiv  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xv  Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii 1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.1  Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  1.2  Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2  1.3  Data examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4  1.4  Relevant practical needs . . . . . . . . . . . . . . . . . . . . . .  7  1.5  Overview of literature . . . . . . . . . . . . . . . . . . . . . . . .  8  1.6  Summary of main concepts . . . . . . . . . . . . . . . . . . . . .  11  1.6.1  Tail order and strength of dependence in the tails . . . . .  11  1.6.2  Tail order and conditional copula . . . . . . . . . . . . .  15  1.6.3  Tail order and Conditional Tail Expectation . . . . . . . .  21  vi  2  3  1.6.4  Tail order and tail order functions . . . . . . . . . . . . .  24  1.6.5  Concordance ordering in the tail . . . . . . . . . . . . . .  26  1.6.6  Sub-extremes and second order conditions . . . . . . . . .  27  1.7  Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  27  1.8  Highlights of selected new results . . . . . . . . . . . . . . . . .  28  Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  29  2.1  Risk measures . . . . . . . . . . . . . . . . . . . . . . . . . . . .  29  2.2  Copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  31  2.3  Regular variation . . . . . . . . . . . . . . . . . . . . . . . . . .  34  2.4  Maximum Domain of Attraction . . . . . . . . . . . . . . . . . .  36  Tail order and intermediate tail dependence . . . . . . . . . . . . . .  38  3.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  38  3.2  Tail orders: definitions and properties . . . . . . . . . . . . . . .  40  3.2.1  Multivariate tail order and tail order functions . . . . . . .  40  3.2.2  Further properties of tail orders . . . . . . . . . . . . . .  49  Intermediate tail dependence : Archimedean copulas . . . . . . .  51  3.3.1  Laplace transform and univariate tail heaviness . . . . . .  52  3.3.2  Upper tail . . . . . . . . . . . . . . . . . . . . . . . . . .  53  3.3.3  Lower tail . . . . . . . . . . . . . . . . . . . . . . . . . .  55  3.3.4  A new parametric Archimedean copula . . . . . . . . . .  57  3.4  Intermediate tail dependence: Mixture of max-id copulas . . . . .  59  3.5  Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  61  3.6  Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  62  Tail comonotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . .  79  4.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  79  4.2  Definitions of tail comonotonicity and properties . . . . . . . . .  80  4.3  Construction of tail comonotonic copulas . . . . . . . . . . . . .  86  4.3.1  Archimedean copulas . . . . . . . . . . . . . . . . . . . .  86  4.3.2  Heavy tail mixtures . . . . . . . . . . . . . . . . . . . . .  89  4.3.3  Extreme value copulas . . . . . . . . . . . . . . . . . . .  93  Asymptotic additivity of risk measures . . . . . . . . . . . . . . .  93  3.3  4  4.4  vii  5  4.5  Concluding remarks and future research . . . . . . . . . . . . . .  98  4.6  Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  99  Tail dependence and conservativity . . . . . . . . . . . . . . . . . . . 111 5.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111  5.2  Conditional specifications . . . . . . . . . . . . . . . . . . . . . . 113  5.3  5.4  6  7  5.2.1  The case: E[X1 |X2 > t] . . . . . . . . . . . . . . . . . . 115  5.2.2  The case: E[X1 |X2 = t] . . . . . . . . . . . . . . . . . . 120  5.2.3  Second order conditions and conservativity . . . . . . . . 121  Asymptotically worst dependence structures . . . . . . . . . . . . 123 5.3.1  Conditional tail expectation . . . . . . . . . . . . . . . . 124  5.3.2  Value at risk . . . . . . . . . . . . . . . . . . . . . . . . . 125  Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.4.1  Conditional specifications . . . . . . . . . . . . . . . . . 126  5.4.2  Asymptotically worst dependence structures . . . . . . . . 127  5.5  Application on a claim dataset . . . . . . . . . . . . . . . . . . . 131  5.6  Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134  5.7  Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136  Second order regular variation and risk measures . . . . . . . . . . 143 6.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143  6.2  Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146  6.3  Univariate cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 148  6.4  Multivariate cases . . . . . . . . . . . . . . . . . . . . . . . . . . 153  6.5  Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . 161  6.6  Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161  Conclusions and future research . . . . . . . . . . . . . . . . . . . . 170 7.1  7.2  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7.1.1  Strength of dependence in the tails . . . . . . . . . . . . . 170  7.1.2  Tail patterns and risk measures . . . . . . . . . . . . . . . 171  Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.2.1  Tail behavior of CTEs . . . . . . . . . . . . . . . . . . . 172  7.2.2  Intermediate tail dependence from hidden regular variation 180 viii  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182  ix  List of Tables Table 1.1  Tail order of various bivariate copulas . . . . . . . . . . . . . .  14  Table 1.2  Tail order and conditional copula . . . . . . . . . . . . . . . .  21  Table 3.1  Tail order of some Archimedean copulas that interpolate independence and comonotonicity . . . . . . . . . . . . . . . . . .  59  Table 5.1  Comparisons between Gumbel, Joe, BB1 and BB7 copula families117  Table 5.2  Condition ψ (ψ −1 (u)) for parametric Archimedean copula families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119  Table 5.3  (More dependence) VaR and CTE for X1 + X2 . The MLEs were based on the whole sample generated from s.BB1 (δ = 1.57, θ = 1.68, λ = 0.77, more dependence) with a sample size of 2000. The bold AIC value is the smallest. . . . . . . . . . . 130  Table 5.4  (Less dependence) VaR and CTE for X1 + X2 . The MLEs were based on the whole sample generated from s.BB1 (δ = 2, θ = 0.4, λ = 0.42, less dependence) with a sample size of 2000. The bold AIC value is the smallest. . . . . . . . . . . . . . . . 131  x  Table 5.5  VaR and CTE for X1 + X2 . The first 4 columns of values are the means of corresponding quantities calculated from 40 random samples generated from the same settings as before. The rest are frequency of those quantities being greater than those for s.BB1.o. (A: more dependent, large sample size; B: less dependent, large sample size; C: more dependent, small sample size; D: less dependent, small sample size. More dependence: δ = 1.57, θ = 1.68, λ = 0.77; less dependence: δ = 2, θ = 0.4, λ = 0.42. Large sample: N = 2000; small sample: N = 50.) . . . . . . . . . . . . . . . . . . . . . . . . 132  Table 5.6  Estimates for margins with/without 1% largest ALAE removed  Table 5.7  Comparison between the s.BB2 and Gumbel copulas: from the  133  second order condition of BB2, we know that θ is the second order parameter that dominates the tail behavior of the copula. If we fit δ, θ of s.BB2 simultaneously for the data, then the MLE of δ tends to be very close to 0; that is, s.MTCJ could better fit the data than s.BB2 (since as δ → 0, BB2 becomes MTCJ). But s.MTCJ will lose the asymptotic full dependence structure. The aim of this comparison is not to find the best fitting copula but to study how conservative s.BB2 is. So we fix two values for the parameter δ for s.BB2 (δ = 0.01, 0.1), and then obtain the MLEs of θ. Table 6.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . 134  How large p must be to get a good second order approximation: the values are the corresponding p for which the absolute difference between the second order approximation and the true value is 5% of the true value. . . . . . . . . . . . . . . . . . . 152  xi  List of Figures Figure 1.1  Data examples - LOSS vs Expense (ALAE) . . . . . . . . . .  5  Figure 1.2  Data examples - Florida flood . . . . . . . . . . . . . . . . .  5  Figure 1.3  Data examples - DJI vs FTSE . . . . . . . . . . . . . . . . .  6  Figure 1.4  Data examples - N225(t) vs N225(t + 1) . . . . . . . . . . .  7  Figure 1.5  Copula is a joint cumulative distribution function (cdf) of uniform margins . . . . . . . . . . . . . . . . . . . . . . . . . .  9  Figure 1.6  Contour plots: Gaussian copula + standard normal margins . .  15  Figure 1.7  Contour plots: Student t copula + standard normal margins . .  16  Figure 1.8  Contour plots: Frank copula + standard normal margins . . .  16  Figure 1.9  Contour plots: Gumbel copula + standard normal margins . .  17  Figure 1.10 Contour plots: MTCJ copula + standard normal margins . . .  17  Figure 1.11 Contour plots: BB2 copula + standard normal margins . . . .  18  Figure 1.12 Tail order and CTE plots of the form E[X1 |X2 > t]. The range of t in each plot was chosen to cover the support of t between the 1% and 95% quantiles. Gumbel(δ = 2), s.Gumbel(δ = 2) and MTCJ(δ = 1). . . . . . . . . . . . . . . . . . . . . . . .  23  Figure 1.13 Tail order and CTE plots of the form E[X1 |X2 = t]. The range of t in each plot was chosen to cover the support of t between the 1% and 95% quantiles. Gumbel(δ = 2), s.Gumbel(δ = 2) and MTCJ(δ = 1). . . . . . . . . . . . . . . . . . . . . . . .  23  Figure 3.1  Contour plots: ACIG copula + standard normal margins . . .  58  Figure 4.1  Florida flood data - Comonotonicity vs Upper comonotonicity  81  xii  Figure 4.2  Simulation of BB2 with/without the univariate margins being transformed to the standard Normal; in the left and middle plots δ = 0.2, θ = 0.4 and in the right plot δ = 0.2, θ = 0.2 .  Figure 4.3  85  Simulation of BB3 with/without the univariate margins being transformed to the standard Normal; in the left and middle plots δ = 0.2, θ = 1.7 and in the right plot δ = 0.2, θ = 1.3 .  Figure 5.1  86  The value of y-axis is E[X1 |X2 > VaRp (X2 )]. In the first plot, X1 , X2 have Exponential distributions with cdf F (x) = 1 − Exp(−x/σ). In the second plot, X1 , X2 have Pareto distributions with cdf F (x) = 1 − (1 + x/σ)−θ . . . . . . . . . . 128  Figure 5.2  The value of y-axis is E[X1 |X2 = VaRp (X2 )]. In the first plot, X1 , X2 have Exponential distributions with cdf F (x) = 1 − Exp(−x/σ). In the second plot, X1 , X2 have Pareto distributions with cdf F (x) = 1 − (1 + x/σ)−θ . . . . . . . . . . 129  Figure 5.3  The value of y-axis is E[ALAE|LOSS ≥ VaRp (LOSS)]. “Empirical” is based on the data with the 1% largest removed; “Empirical0” is based on the original data. . . . . . . . . . . . . . 135  Figure 6.1  Sub-extremal relationships between CTE and VaR (α = 2) for Hall/Weiss class . . . . . . . . . . . . . . . . . . . . . . . . 151  Figure 6.2  First/Second order approximations for Burr scale mixture of normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159  Figure 7.1  CTE plots for E[X1 |X2 > t]. The range of t in each plot was chosen to cover the support of t between the 1% and 99% quantiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174  Figure 7.2  CTE plots for E[X1 |X2 = t]. The range of t in each plot was chosen to cover the support of t between the 1% and 99% quantiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179  xiii  Glossary cdf cumulative distribution function ACIG Archimedean Copula based on the Laplace Transform of Inverse Gamma MTCJ Mardia-Takahasi-Cook-Johnson copula MRV Multivariate Regular Variation HRV Hidden Regular Variation 2RV Second Order Regular Variation CTE Conditional Tail Expectation VaR Value at Risk MDA Maximum Domain of Attraction ALAE Allocated Loss Adjustment Expenses LOSS Indemnity Payment DJI Dow Jones Industrial Average Index FTSE FTSE 100 Index N225 NIKKEI 225 Index AMA Advanced Measurement Approaches LT Laplace Transform  xiv  Acknowledgments I am greatly indebted to my supervisor Professor Harry Joe. Professor Joe is an encyclopedia to me. Every time when I asked questions, I can get prompt and instructive suggestions from him. Professor Joe has written up many inspiring notes to motivate research questions, and to help me develop my own research skills; I feel that he cares about my research and even my future careers very much. He masters the art of supervision: he gave me sufficient pressure to move forward, and also allowed me to adjust my own pace to it. With the excellent supervision of Harry, I, just like a baby who starts learning to walk, have accumulated great encouragement to step forward, because I know that a powerful arm is just around and is always ready to pull me up. I would like to express my sincere thanks to Professors Haijun Li, Natalia Nolde and Mat´ıas Salibi´a-Barrera, who are members of the Comprehensive Committee and/or the Supervision Committee. Professor Li is very warm-hearted and he has pointed out several interesting topics to motivate my research interests. Special thanks are given to Natalia; she helped me a lot for improving my writing, not only for the thesis but also for other application documents. I also want to thank Professor Christian Genest for acting as the external examiner, and Professors Edwin Perkins and Ruben Zamar for serving as the university examiners. I want to extend my sincere gratitude to the faculty and staff of the department, to those who have directly or indirectly helped me during my study at UBC. I have taken two courses in statistical consulting with Professor John Petkau, who helped me a lot in improving my statistical thinking and writing. I also want to say Thank You to Peggy Ng, Elaine Salameh, Andrea Sollberger, and Viena Tran for lots of prompt helps and routine work. xv  I also thank the department Ping Pang players Professors Ruben Zamar and Lang Wu, and Chen Xu, Xu Wang and Yongliang Zhai, for many fun times, and thank the other fellow students who have helped me during these enjoyable four years. Special thanks to Professor Ka Chun Cheung, my Master’s supervisor, and the Department of Statistics and Actuarial Science at the University of Hong Kong; I was preparing the thesis proposal with the kind hospitality of Ka Chun and the financial support from Canada Graduate Scholarships - Michael Smith Foreign Study Supplements Program. This thesis has been written up with the generous financial support from the Natural Sciences and Engineering Research Council of Canada, to which I want to express my sincere thankfulness. Lastly and most importantly, I sincerely thank my wife, Chaoxiong Xia, for her love, encouragement and support in every moment of my life, and my two little kids Claire and Daniel, for bringing me so much happiness during the journey of completing the thesis.  xvi  Dedication  xvii  Chapter 1  Introduction 1.1  Motivation “The interviewed firms described some general common lessons learned from the crisis, of which probably the most commonly mentioned was the necessity (but difficulty) of capturing tail risks and dependencies resulting from tail events.” Developments in Modelling Risk Aggregation Basel Committee on Banking Supervision, 2010  The research activities on dependence modeling of large losses and their risk assessment have been sparked by the contagion of the global financial crisis arising in the year of 2007. There have been tons of articles debating the roots of the crisis from many perspectives. One common opinion from the viewpoint of researchers in quantitative fields is that the high-risk scenarios as well as their comovements in the global financial system have been ignored, which hides the high possibility of extremal losses that could happen simultaneously. This thesis is not going to join the discussion, but attempts to contribute to the understanding of quantitative and/or statistical methodologies that account for the phenomenon of heavy tails of each individual risk and their dependence as well. Among many issues of modeling insurance losses or financial returns, estimating, predicting and pricing potential extremal risks are particularly important. 1  Moreover, many extremal losses have been observed to have occurred consecutively or simultaneously. The comovement, in the meanwhile, could considerably increase the entire risk. Thus, the dependence of extremal risks is expected to play an important role for integrated risk management. Although the statistical methodologies for studying overall properties of random samples such as means are relatively mature, sound methodologies for studying the dependence relationship between high quantities are far from being well-developed.  1.2  Notation  The following notation will be used throughout the thesis. We use bold letters for vectors, such as x := (x1 , . . . , xd )T , where T is the transpose operation of a matrix. 1d = (1, . . . , 1) with d elements, and then u1d = (u, . . . , u); Id = {1, . . . , d}. N+ = {1, 2, . . . }; R := (−∞, ∞); R+ := [0, ∞); R+ := [0, ∞]; R := [−∞, ∞]; A ⊂ B means that A is a subset (not necessarily a proper subset) of B; Cartesian product is denoted as [a, b] := [a1 , b1 ] × · · · × [ad , bd ]; let x := (x1 , . . . , xd ) and y := (y1 , . . . , yd ), then x ≤ y means that xi ≤ yi for all i = 1, . . . , d. [x] = max{t integer : t ≤ x}; ψ (i) (s) is the ith order derivative of ψ evaluated at s; 1A (x) is the indicator function such that 1A (x) = 1 if x ∈ A, and 1A (x) = 0 if x ∈ / A. For a random variable X, the generalized inverse of its cumulative distribution function (cdf) is defined as FX−1 (p) = inf{x ∈ R : FX (x) ≥ p}. Also, note that for all x ∈ R and p ∈ [0, 1], we have (e.g., Dhaene et al., 2002a) FX−1 (p) ≤ x ⇐⇒ p ≤ FX (x).  (1.1)  A random variable is said to be continuous if its cdf is continuous. The survival function is F j = 1 − Fj if Fj is a cdf. For a d-dimensional multivariate cdf F , the survival function is F = 1 + (−1)|I|  ∅=I⊂{1,...,d} FI ,  where FI is the cdf for the  d  I-margin. X = Y means X and Y equal in distribution. For a bivariate differentiable cdf F , we write (∂1 F )(x1 , x2 ) := ∂F (x1 , x2 )/∂x1 = F2|1 (x2 |x1 )f1 (x1 ),  2  where f1 = F1 . Unless otherwise mentioned, a distribution is assumed to be absolutely continuous with respect to Lebesgue measure. Throughout the thesis, we use R(F1 , . . . , Fd ) to represent the Fr´echet space of all d-dimensional random vectors with F1 , . . . , Fd as the marginal distributions. Tν (·) is the cdf of the standard univariate Student t distribution with degree of freedom ν, and tν (·) is the corresponding density function. Tν,Σ (·) is the cdf of the standard multivariate Student t distribution with degree of freedom ν and covariance matrix Σ, and tν,Σ (·) is the corresponding density function. Φ(·) is the cdf of the standard univariate Normal distribution and φ(·) is the density function. For the standard multivariate Normal distribution, the cdf and density function are ΦΣ and φΣ , respectively. Throughout this thesis,  will be used to represent a  correlation coefficient and ρ is reserved for the second order parameter of Second Order Regular Variation (2RV). The notation C will always represent a copula function, a multivariate cdf with U (0, 1) univariate margins. The conditional distributions of a bivariate copula are written as C1|2 (u|v) = ∂C(u, v)/∂v and C2|1 (v|u) = ∂C(u, v)/∂u. A density function of copula is written as c(u, v) := ∂ 2 C(u, v)/∂u∂v. Second order partial derivatives are D22 C(u, v) = ∂ 2 C(u, v)/∂ 2 v. For a copula, say, the Gumbel copula, the corresponding survival copula is denoted as “s.Gumbel”; that is,“s.” symbolizes the survival copula. A lower tail dependence parameter is denoted by λL , and an upper tail dependence parameter is denoted by λU . We use κ for tail order, and κL and κU are lower and upper tail orders, respectively. For positive functions f, g, asymptotic equivalence is denoted as f ∼ g if g if lim inf f /g ≥ 1, and asymptotic  lim f /g = 1, asymptotic inequality f inequality f  g if lim sup f /g ≤ 1. The notation f (t)  g(t), t → ∞ means  that f is ultimately greater than g; that is, ∃t0 such that t ≥ t0 implies that f (t) ≥ g(t). h(t) = o(g(t)) means limt→t0 h(t)/g(t) = 0, and h(t) = O(g(t)) means 0 ≤ limt→t0 h(t)/g(t) < ∞, where −∞ ≤ t0 ≤ ∞ is a given limiting point. The notation RVα represents the class of functions that are regularly varying at ∞ with index α ∈ R, and RVα (0+ ) represents the class of functions that are regularly varying at 0 with index α ∈ R. MRVd (α) represents the class of ddimensional multivariate regularly varying random vectors with α ∈ R the regular 3  variation index. (x) is used as a slowly varying function. The letter E represents mathematical expectation, and Var is for variance. Note that the Var here is different than the notation VaR (Value at Risk) defined in Definition 2.1; the latter is used in insurance and finance. x ∨ y := max{x, y}; x ∧ y := min{x, y}.  1.3  Data examples  In this subsection, we will present some data examples to motivate various patterns of tail dependence and tail asymmetry. The marginal distributions are usually transformed to a standard Normal distribution to visualize the patterns of upper and lower tails. Let us look at a well known dataset in the actuarial literature (Frees and Valdez, 1998). This dataset comprises 1500 general liability claims with Indemnity Payment (LOSS) representing the amount of payment and Allocated Loss Adjustment Expenses (ALAE) that are specifically attributable to each claim and may include legal expenses, investigation expenses, etc. A typical pattern for this dataset is that the margins are heavy-tailed and the degree of positive dependence between large values is larger than that between small values. This kind of tail asymmetry pattern is not clear as one looks at the scatter plot for the original data (left panel of Figure 1.1). In order to visualize the asymmetric tail dependence, we usual transform each margin to be distributed as univariate standard normal, and then look at the scatter plot for the transformed pairs (right panel of Figure 1.1); such transformed values are referred to as normal scores. A stronger upper tail dependence for this dataset appears, and it makes sense since a claim with a larger loss amount tends to involve more legal expenses and demands more time and costs for investigation. The asymmetric tail dependence pattern can also be observed in environmental data, such as a Florida flood loss dataset. There are 67 counties in Florida. The dataset comprises the monthly amount of losses for each county from 1977 to 2006, and it is a part of the Spatial Hazard Events and Losses Database for the United States (SHELDUS) that is maintained by the Hazards and Vulnerability Research Institution. We choose randomly 30 counties and add the loss amount among these counties together to get a monthly aggregate loss for these counties. Then we  4  4  6e+05  Figure 1.1: Data examples - LOSS vs Expense (ALAE)  ● ● ● ● ● ● ● ●●● ● ● ●● ● ●● ● ● ●● ● ● ● ● ●● ● ● ●● ●●● ● ●● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ● ●●● ●● ● ● ●● ● ● ●● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ●● ●●● ● ●●● ● ●● ●● ● ●● ● ● ● ● ●● ●● ●● ● ● ● ●●● ● ● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ●● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ●●● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ●●● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ● ●● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ●●●● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●● ●● ●● ● ● ● ● ●● ● ● ● ●● ● ●● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●●● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ●● ● ● ● ●● ●●●●● ●● ●●●● ● ●● ●● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ●● ●● ●● ●● ● ● ●●●●● ● ● ●●● ● ● ●● ●● ●●● ● ●●● ● ● ● ●● ●● ●● ● ● ● ● ●●● ●●●● ● ● ● ● ● ● ●●●● ●●●● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ●● ●●● ●● ●●●●● ● ● ●● ● ●● ●● ● ● ● ● ●● ●●●●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ●  ●  0  2 −2  ●  ●  ●  ●  ●●  500000  ●  −4  2e+05  ●  ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●●● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●  ●  0  Norm(ALAE)  4e+05  ●  0e+00  ALAE  ●  1500000  2500000  −4  −2  LOSS  0  2  4  Norm(LOSS)  randomly choose another 30 counties from the rest counties to get the monthly aggregate loss similarly. The dependence between these two aggregate losses can be demonstrated by Figure 1.2. It is very clear that the upper tail dependence is stronger than the lower tail dependence. This phenomenon of stronger upper tail dependence for aggregate losses is related to a concept of micro-correlation that is studied in Cooke et al. (2011). Figure 1.2: Data examples - Florida flood  Normal score of aggregate loss of other 30 counties  Florida flood aggregate loss (1977~2006)  ● ●  2  1  0  −1  −2  ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● ● ●●●●●●●● ●● ● ● ● ● ● ● ●● ● ●●● ● ● ●●● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●●●● ● ● ● ●●●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ● ●● ● ● ●● ●● ●● ●● ● ● ●● ●● ●● ● ● ●●● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●  ●  ●  −2  −1  0  1  2  Normal score of aggregate loss of 30 counties  5  In the natural world or in our human society, there seems to be certain mechanism that drives the happening of clusters of high-risk events. Not only in insurance practice, but also in the financial market, we may observe such tail asymmetry phenomena. Figure 1.3 is a scatter plot of normal scores for weekly log returns between Dow Jones Industrial Average Index (DJI) and FTSE 100 Index (FTSE) during the period of October 1st, 2005 to June 30th, 2007. The plot illustrates that the dependence between large losses of two stock markets is higher than the dependence between large returns. Figure 1.3: Data examples - DJI vs FTSE DJI vs FTSE (20051001~20070630, weekly)  Normal score of weekly logreturn of FTSE  ● ●  2  ● ●  ● ●  ●  1  ● ●  ●  ● ●●  ● ●  0  ● ●  ●●  ●  ● ●  ●  ●  ●● ● ● ●● ●  ● ●  ●  ● ●  ●  ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ●● ● ● ● ●● ● ● ● ●  −1  ●  ● ● ● ● ●● ●  ● ●  ● ● ●  ● ●  ●  ● ●  ● ● ●  −2  ● ●  −2  −1  0  1  2  Normal score of weekly logreturn of DJI  Serial dependence within a univariate financial time series may also appear to be tail asymmetric. For example, if we take the squared daily log returns as approximations to observed volatilities, then the asymmetric pattern of serial dependence between such observed volatilities can be illustrated by Figure 1.4. Here we use NIKKEI 225 Index (N225) during the period of October 1st, 2005 to March 9th, 2009 as an example. The observed volatilities tend to clutter together, which has already been noticed by many researchers. We refer to Engle (2004), written by one of the two holders of Nobel Prize of 2003 in economics, for an excellent review on volatility clustering. Note that the above scatter plots are simply used to visualize the asymmetric tail dependence patterns; those are not necessarily meaning that certain bivariate  6  Figure 1.4: Data examples - N225(t) vs N225(t + 1) NIKKEI 225 (20051001~20090309, daily)  ●  Normal score of squared logreturn at t+1  3  ● ● ● ● ● ● ● ●● ● ●●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●●● ● ●● ● ● ● ● ● ● ● ●●●● ● ●● ●● ● ● ● ● ●● ● ● ●● ●● ● ●●●● ●● ● ●● ● ●●● ● ● ●●● ● ● ●● ● ● ● ●●● ● ● ●●● ●●● ● ● ● ● ●● ● ●● ● ● ●●●●●● ● ●● ●● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●●●● ●●● ●● ●●●●● ● ● ●● ● ● ●● ●● ●● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ●● ●● ● ●●●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ●●● ●● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ● ● ●● ● ● ●● ● ● ●●●●●●● ● ●● ● ●● ● ● ● ● ● ● ●●● ● ●●● ● ●●● ● ●● ●● ● ● ● ●●● ●● ● ● ● ● ● ●● ●● ●● ●● ● ●● ● ●●●●● ● ● ●●● ● ●● ● ●● ● ●●●● ●● ● ●●● ● ● ● ●● ●● ● ● ●● ● ●● ●●● ● ●● ● ● ● ● ● ●● ● ●● ● ●● ●●● ● ● ● ● ● ● ● ●●● ● ●●●●●● ● ●●● ● ● ● ● ●● ●● ● ●● ● ●● ● ●● ● ● ● ● ●● ●● ●●● ● ●●●● ●● ●● ●● ●●● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●●● ● ●●● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ●●● ● ● ● ●●● ● ● ●● ● ● ● ●●●● ● ●● ● ● ●●● ●● ● ●● ● ● ●● ●● ● ● ●● ● ● ●●● ●● ● ●●● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ●●● ●● ● ● ●●● ● ● ●● ● ●● ●● ● ●● ●● ●● ● ● ● ● ●● ● ● ●●●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ● ● ● ●●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●  ●  ●  ●●  ●  2  1  0  −1  −2  −3  −3  −2  −1  0  1  2  3  Normal score of squared logreturn at t  distributions are ready to be used for fitting those data (or transformed data), as the data points in the plots may not even be a random sample. For specific questions, one needs to fully consider the dependence structures that saturate the data of interest to decide where a dependence model such as a copula can be appropriately applied.  1.4  Relevant practical needs  The term of tail modeling in this thesis is referred to as statistical modeling for marginal distributional tails and their dependence structure in the tails of the joint distribution. Traditional areas that require tail modeling include rate makings for reinsurance, auto insurance, etc., and statistical modeling in finance and various subject scientific fields such as environmental science and Internet engineering. There are also some emerging needs for tail modeling in quantitative risk management for the bank and insurance sectors. For instance, under Basel III – a global regulatory standard on bank capital adequacy, stress testing and market liquidity risks – operational risk charges can be calculated based on the so-called Advanced Measurement Approaches (AMA). Operational risk is typically very heavy-tailed (Dutta and Perry, 2006), and the modeling and risk assessment is actually very challeng7  ing (Chavez-Demoulin et al., 2006; Cope et al., 2009). In parallel, under Solvency II, there are relevant regulatory requirements on operational risks for insurers. First of all, AMA itself is far from being well developed. Sound methodologies/strategies for risk assessment and management are in high demand. Secondly, the developing market of operational risk mitigation products would entail huge potential for research on tail modeling; for example, how to conduct tail risk securitization and how to design and price relevant products?  1.5  Overview of literature  Before dealing with any particular models, a fundamental question is how to model the dependence between various univariate random variables, especially for those with non-Gaussian distributions. In the past practice of actuarial science, quantitative finance and so on, dependence structures between random variables were often studied with some summary quantities, such as, correlation coefficient, Kendall’s τ and Spearman’s ρ. Especially the usage of correlation coefficient has been popular. It is well known that the maximum correlation of 1 will be reached if and only if the two random variables have the same distribution up to a linear transformation (Rodgers and Nicewander, 1988). Shortcomings of using correlation coefficient to model the dependence for insurance and finance have been observed by many papers. We refer to Embrechts et al. (2002) for an inspiring reading. Apparently, every single summary statistic is not able to fully describe the dependence structure between random variables. Instead, the copula approach becomes more and more popular for modeling dependence structures of insurance losses or financial returns where marginal distributions are often heavy-tailed and skewed. We list here, among many of others, some advantages of copula approaches: (1) the copula itself is a joint cdf, which captures much more information than any single quantity; (2) when the univariate margins are all continuous, the copula function can be uniquely determined, thus fully characterizing the dependence structure; (3) as a multivariate cdf, copula can be parametrized and is useful for statistical modeling and inference; (4) the copula provides a great deal of flexibility for constructing multivariate distributions. The copula itself is a joint cdf with Uniform(0, 1) marginal distributions. Let  8  F be the joint cdf, F1 , . . . , Fd be the univariate cdfs, and C be a copula, then F can be written as F (x1 , . . . , xd ) = C(F1 (x1 ), . . . , Fd (xd )). For the insurance loss data in Figure 1.1, if we transform each marginal data via the corresponding empirical distribution to get data that is approximately uniformly distributed on [0, 1], then the scatter plot for the pairs of transformed data is Figure 1.5. Roughly speaking, the copula C can be used to fit such transformed data. A more formal introduction to copulas is in Section 2.2.  0.6 0.4 0.0  0.2  Unif(ALAE)  0.8  1.0  Figure 1.5: Copula is a joint cdf of uniform margins  ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ●● ●●● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●●● ●● ● ● ●● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ●●●● ● ●●●●● ● ● ●●● ● ● ● ●● ● ● ● ●● ● ● ●● ●●● ● ●● ● ●●●● ●● ● ● ●●● ● ●● ●● ●● ● ● ● ●● ● ●● ● ●●● ● ● ● ● ● ● ●●●● ●●● ● ●● ●● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ●●●●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●●● ● ● ● ● ● ●● ●●● ● ● ●●●● ● ●● ●●●● ●● ●● ● ● ●● ● ●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ●● ●●● ● ● ●● ● ● ● ● ●● ●● ● ● ● ●● ● ● ●●● ● ● ●● ● ●● ●● ● ● ● ●● ● ●● ● ●● ●● ●● ● ● ● ●● ● ●● ●●●● ●●● ● ● ●●●● ●● ● ●● ●● ● ● ●●● ● ● ● ●● ● ● ● ●● ● ●●●●● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ●● ●● ●●●● ●● ● ●● ●● ● ● ●●● ● ●●● ● ●●● ● ● ● ● ● ● ● ●●● ● ●● ●● ● ● ● ●● ● ●● ●●● ●● ● ● ●●●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ●●● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●●● ● ● ● ● ● ●● ● ●● ●●● ● ● ● ● ● ●●● ● ●● ●●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●●●●● ● ● ●●● ● ● ● ●● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ●● ● ●● ● ●●● ●● ●● ● ● ●● ● ● ●● ● ● ● ●●● ●●●● ●●● ●●●●● ●● ●●● ● ● ● ●● ● ●● ● ●● ● ●●●● ● ●●● ●●● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ● ●● ● ●●● ● ● ●● ●● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●●●●● ● ● ●● ● ● ●● ● ● ●●● ● ●● ● ●● ●● ●●● ● ● ●●● ● ●● ● ●● ● ● ● ●●● ●●●● ● ● ●●● ● ● ●●●● ● ● ●● ●● ●● ● ●● ●● ●● ● ●●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●●● ●● ● ●● ● ●● ●● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ●●● ● ● ●● ● ● ●●●● ●● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ●● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●●●●● ●● ●●● ● ● ● ●● ●● ● ● ● ●● ●● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ●● ● ● ●● ●●● ● ● ●●●● ● ●● ● ● ● ●● ● ● ●● ● ●● ●● ● ●● ● ● ●● ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ●● ● ●● ●●● ● ● ● ● ● ●●● ●● ● ● ●●● ●● ● ● ● ●●● ● ●● ●● ● ● ● ● ● ● ●●● ●● ●●● ● ●●● ●● ● ● ● ● ●● ●● ●● ● ● ● ● ● ● ●●●● ● ● ●● ● ●● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ● ●●●● ● ●●●●●● ● ● ● ● ● ● ●● ●● ● ● ● ●●● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●●●●● ●● ● ●● ●● ● ●●●● ●●●● ● ● ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●  ●●  0.0  0.2  0.4  0.6  0.8  1.0  Unif(LOSS)  A pioneer paper on copula theory is Sklar (1959), which established the well known Sklar’s theorem: there is a copula function for joint distributions and the copula is unique if the univariate marginal distributions are continuous. Standard references for dependence modeling and copulas are Joe (1997) and Nelsen (2006). The former is a very informative monograph on dependence concepts and relevant statistical modelings, where copula is one fundamental tool for statistical dependence modeling. The latter deals with the concept of copula in a more mathematical way, and focuses on relevant theories of copulas. For contributions to the classic statistical issues on copula modeling, such as statistical inference and goodness-offit tests, we refer the reader to the following contributions and references therein: 9  Genest and Rivest (1993), Genest and Favre (2007), Genest et al. (2009b) and Brechmann et al. (2012). Research activities on copula modeling in actuarial science and quantitative finance have sparked since Frees and Valdez (1998), and Li (2000). Two excellent reference books on dependence modeling in actuarial science and quantitative risk management are Denuit et al. (2005) and McNeil et al. (2005). Most recently, Kurowicka and Joe (2011) and Jaworski et al. (2010) have provided good summaries of copula theories and applications: the former promotes a relatively new concept of vine copulas, and the latter contains several survey papers on copulas. The copula approach is particularly useful for studying tail behavior of multivariate non-Gaussian distributions (Joe, 2006; Mikosch, 2006). It is extremely flexible to use copula to construct multivariate distributions that are able to account for various distributional tails. In the practice of actuarial science and quantitative risk management, a more relevant task is to model and conduct risk assessment for the joint tails of dependent losses, especially for large or extremal risks. The classic univariate extreme value theory provides a sound theoretical ground for building up useful approaches for assessing univariate marginal tails. Compared with the univariate extreme value theory, the multivariate extreme value theory has not been well developed. In parallel with the copula approach, researchers find that transforming the marginal distributions to those having the same power law can be more convenient in some situations. This approach is referred to as Multivariate Regular Variation (MRV), and its root, the theory of univariate regular variation has been nicely developed. There are some excellent reference books for extreme value theory, and for regular variation. de Haan and Ferreira (2006) is a nice monograph on theoretical aspects of extreme value theory and includes some discussion on 2RV, Embrechts et al. (1997) has become a classic for modeling univariate extremal events, and Coles (2001) provides a clear and more intuitive introduction to statistical modeling of univariate extreme values. The books Beirlant et al. (2004) and Reiss and Thomas (1997) comprise many interesting topics for statistical modeling of extreme values that include not only univariate but also multivariate extreme values, and Falk et al. (2010) is a comprehensive monograph on theories and applications of multivariate extreme values such as multivariate Peaks Over Threshold (POT) 10  approaches. For researchers who study the theory of regular variation and who may use regular variation as a tool, Bingham et al. (1987) should be a must-have and it provides encyclopedic reference on the theory of univariate regular variation. For MRV, Resnick (1987) and Resnick (2007) are standard reference books, and the latter also reflects some recent development of MRV. The above literature review is about the general idea of copula theory and extreme value theory. More specific literature reviews for concrete topics that belong to these research themes will be conducted in later chapters when it becomes necessary.  1.6  Summary of main concepts  In this section, we will highlight different concepts for copulas and their joint tails; this is to emphasize that different copula families have different tail properties which may have a large effect on inferences such as joint tail probabilities (chance of simultaneously large losses from different risks) and conditional tail expectations. The bivariate case will be employed here to explain the basic ideas, and those ideas can be extended to the multivariate case.  1.6.1  Tail order and strength of dependence in the tails  One theme of the thesis is to study the strength of dependence in the tails. In statistical modeling, a basic idea is to find a summary quantity that can capture useful information of interest. Such a quantity should be simple and be able to account for as much as possible relevant and critical information. If we use a copula C to model the dependence between two continuous random variables X and Y , then the graph of the map u → C(u, u), u ∈ [0, 1] describes the evolution of joint probabilities of X and Y . In order to study the tail behavior of the map and thus the tail dependence between X and Y , we hope to have a certain simple function g that can be used to approximate C(u, u) as u → 0+ ; that is, C(u, u) ∼ g(u), u → 0+ . Such a function g(u) should be bounded by u and go to 0 as u → 0+ . Therefore, a suitable class of functions for g can be the class of regularly varying functions with the minimum regular variation index being 1, e.g., g(u) = uκ (u) where 1 ≤ κ and is a slowly varying function such as (u) = (− log u)−1/3 . A formal 11  introduction to regular variation is in Section 2.3. For the function g, the term uκ dominates the behavior of g(u) as u → 0+ . So we choose κ as such a summary quantity that can be used to capture the information of tail association patterns, and κ is referred to as tail order. A smaller κ means higher joint probability in the lower corner as u → 0. Although obviously a single quantity such as the tail order κ can not capture all the information on the tail, it provides relevant and critical information of interest, especially when we compare tail orders of copulas within the same copula family, or when the information of interest is for a sufficiently small u. Therefore, such a simple quantity may also be useful in statistical modeling. When κ = 1, that is, C(u, u) ∼ u (u), u → 0+ , and limu→0+ (u) = λ, if 0 < λ ≤ 1, it is said that the copula C has lower tail dependence and the limit λ (if exists) is referred to the tail dependence parameter (some papers use “tail dependence coefficient” for the same meaning). The concept of tail dependence has been studied intensively in the literature, e.g., Joe (1997) and Schmidt (2002). As a special case of the usual tail dependence, the so-called tail comonotonicity corresponds to that the tail order κ = 1 and the tail order parameter λ = 1 as well. Tail comonotonicity is the strongest tail dependence structure. In situations where no sufficient data in the upper tail of certain joint distributions is available and the statistical inference of interest is aimed at the region beyond a very high threshold, which is often the situation that one meets in quantitative risk management, tail comonotonicity may provide a conservative dependence structure. When 1 < κ < 2, we refer to this case as intermediate tail dependence, and the limit (if exists) of the slowly varying function is referred to as the tail order parameter. When κ = 2 and the limit (if exists) of the slowly varying function is finite and nonzero, then we use the notion of tail quadrant independence (Reiss, 1989). For d-dimensional cases with κ = d > 2, we employ the term of tail orthant independence. Note that, in the literature of the extreme value theory, terms of “asymptotic independence” or “tail independence” have often been used to represent the cases where κ > 1. However, in this thesis, in order to discriminate the two cases of 1 < κ < 2 and κ = 2, we use the notation of “intermediate tail dependence” and 12  “tail quadrant independence”, respectively. Intermediate tail dependence for the multivariate case corresponds to 1 < κ < d. Upper tails can be dealt with similarly: the upper tail behavior of copula C(u, v) is simply the lower tail behavior of its survival copula C(u, v) := u + v − 1 + C(1 − u, 1 − v). If C(u, u) ∼ uκ (u), u → 0+ , then the κ here is referred to as the upper tail order of the copula C. For any of the above cases, the limit of the slowly varying function  can be  referred to as the tail order parameter no matter it is the case of tail dependence, intermediate tail dependence or tail quadrant independence. For comparing the strength of dependence in the tails, we first compare tail orders that dominate the tail dependence pattern. If tail orders are the same, then tail order parameters can be used. Examples of lower and upper tail orders for commonly used Archimedean copulas are listed in Table 1.1, and the contour plots of these copulas are given in Figures 1.6, 1.7, 1.8, 1.9, 1.10 and 1.11. Note that, the first three examples of Gaussian, Student t and Frank copulas are all reflection symmetric; that is, the pattern of the upper tail is the same as the pattern of the lower tail. But tail asymmetry appears in the contour plots of the other three examples. When tail order κ > 1, such as the tails of Gaussian copula and the lower tail of the Gumbel, the contour plots look more round than the contour plots for tails with κ = 1. We refer to Balkema and Nolde (2010) for sufficient conditions of κ > 1 from a geometric perspective.  13  Table 1.1: Tail order of various bivariate copulas  14  Copula Gaussian Student t Frank  cdf ΦΣ Φ−1 (u), Φ−1 (v) ; Σ : correlation matrix, Φ : cdf of normal Tν,Σ Tν−1 (u), Tν−1 (v) ; 0 < ν, Σ : correlation matrix, Tν : cdf of Student t − log [1 − e−δ − (1 − e−δu )(1 − e−δv )]/(1 − e−δ ) /δ; 0 ≤ δ  Gumbel  exp − (− log u)δ + (− log v)δ  MTCJ BB2  u−δ + v −δ − 1  −1/δ  C = 1 + δ −1 log e  1/δ  ;1 ≤ δ  ;0 ≤ δ  δ(u−θ −1)  δ(v −θ −1)  +e  −1  κ of C(u, v) Lower Upper  κ of c(u, v) Lower Upper  2 1+  2 1+  −2 1+  −2 1+  1 2  1 2  -1 0  -1 0  21/δ  1  21/δ -2  -1  1  2  -1  0  1  2  -1  0  −1/θ  ; 0 < θ, 0 < δ  When the tail order for the copula density function is negative such as Gaussian copula (Table 1.1), the density at the boundaries (0, 0) and (1, 1) of the copula is infinity. The quantities of tail order and tail order parameters only capture the tail behavior along the diagonal of the copula. We may use a function b to obtain more information about limiting behavior of copulas, namely, lim C(uw1 , uw2 )/uκ (u) =: b(w1 , w2 ),  u→0+  w1 , w2 > 0.  The b function is referred to as the tail order function of the copula C. For detailed study on tail order and tail order functions, we refer the reader to Chapter 3. Figure 1.6: Contour plots: Gaussian copula + standard normal margins  2  ρ = 0.4  2  ρ = 0.2  2  ρ=0  0.02  0.06  4 0.1  0.1  2  −2  −1  0  1  2  −2  −1  0  1  2 0.02  0.08  −1  0  1  2  0.18  0.12  0  6  6  0.0  −2  0.02  −2  −2  −2  1.6.2  0.0  0.04  −1  6  −1  −1  0.0  0.02  8 0.0  0.12  0  0  0.  1  2  0.1  0.04  1  1  1  0.04 8 0 .1  0 0.  2  ρ = 0.8  2  ρ = 0.6  2  ρ = 0.5  0.1  1  −2  −2  0  0.12  0  −1  0.04  0.02  −2  −1  −1  0  0.08  0  0.1  0.08  2 0.1 6 0.0  0.1  0.04  −2  6 0.0 8  0.0  0.12  −1  1  0.04  1  1  0.02  −2  −1  0  1  2  −2  −1  0  1  2  Tail order and conditional copula  Now let us look at how tail order will affect conditional copulas of the form C1|2 (u|v) as v goes to 0 or 1. We use C1|2 (u|0) and C1|2 (u|1) to represent these 15  Figure 1.7: Contour plots: Student t copula + standard normal margins  0.02  0.02 0.06  12 0.  2  −2  0  1  1  0.1  0.16  8  25  0.2  −1  −1  2  5  0  4  0.  0.0  0.1  −2  −2  −2  1  2  0.05  0  0  1  2  2 0.2  0.1 6  0.14  0  1 0  0  0.1  .06  −1  −1  ρ = 0.8, df = 2  0.1  0.06  −2  −2  2 0.1  2 0.1  −1  2  0.04  0.02  0.08  0.1  25 0.  −1 −2  −1  0.04  0.02  0.05  ρ = 0.6, df = 2  2  ρ = 0.4, df = 2  5  0.1  1  1  0  14  8  0.0 0.04  −2  −2  0  0.  0.16  0.08  −1  0.2  0 −1  −1  0. 14  0  2  0.04  −2  1  0.1  0.1  0.1  1  1  0.06 1 0.  ρ = 0.8, df = 4 2  ρ = 0.6, df = 4 2  2  ρ = 0.4, df = 4  −2  −1  0  1  2  −2  −1  0  1  2  Figure 1.8: Contour plots: Frank copula + standard normal margins δ = 0.8  0.04  2 1  1  0.1  0  0  0  4 0.1  2  0.08  0.06  0.1  −1  −1  0.08  −2  −1  0  1  2  −2  0.04  −2  −2  0.04 0.02  −2  −1  0  1  2  −1  0  0.04 8  0.0  1  8 0. 1  4 0.1  6  0  16  −1  1 0.  0  1  2  0.06  −2  −2  −2  2  0.06  0.02  −1  0.  0.1  −1  −1  0.1  0.  0. 14  0  0.14  0  12  0.06  −2  2  2  2 1  0.0  2  0.1  1  δ=6 0.04  0.04 0.08  1  −2  δ=4  2  δ=2  0.02  −2  −1  0. 1  −1  4  0.1  2  12  0.  1  0.1  0.1  0.14 0.  0.02 0.06  0.06  0.08  1  δ = 1.6  0.02  2  2  δ = 0.2  0  16  1  2  0.02  −2  −1  0  1  2  Figure 1.9: Contour plots: Gumbel copula + standard normal margins  0.02  2  δ = 1.6  2  δ = 1.4  2  δ = 1.2  0.04  0.04  0.  −1  −1  1  2  −2  −2  −2  0  −2  −1  0  1  5  2  1  0.  3 0.  0  6  0.  0  0. 1  −1  25 0.  0.1  1  2  0.02  −2  0.05  −2  −2  0.06  0.02  2  0.1  2 0.1  −1  0.1  0  1  14  0.16  8  0  0.1  4 .1  0.06  −1  0  2  0.04  8 0.0  0  −1  −1  0.1  1  1  2  −2  δ=4  2  2  0.04  0.1  2  0.02  δ=2  8 0.0  −2  0.12 0.06  0.02  δ = 1.8  −2  0. 14  0  0. 14  16  0  0.1  0 −1  0.12  0.06  0.04  −1  6  12 0.  0.08  −2  0.08  0.1 0.1  4  0.1  0.08  0.1  1  1  1  0.06  −1  0  1  2  −2  −1  0  1  2  Figure 1.10: Contour plots: MTCJ copula + standard normal margins δ = 0.8  0.04  1  0.08  1  1  2  −2  0  1  −1  0  2  2  0.02 0.06  1  0.12  0  1  δ = 1.4  0.12  0  1  14  0.  0.16  4  −2  0.06  0.12 0.1  2  0.02  2  0.02  2  0.04  δ = 1.2  0.06  1  0.08  −2  −1  0.16  −2  −2  0  0.02  δ=1  0  0.1  −1  −1  4 0.1 0.1  0.06  0.02  −1  0. 16  0  0  0 −1  0.1  0.06  −2  0.12  2  1 0.  0.14  16 0.  −1  1  0.08  0.06  0.14  0.04  0.12  0.02  2  2  δ = 0.2  2  δ = 0.1  0.1  14  0.  −1  0.08  −2  8  0.0  −1  0  1  2  −2  −1  8  0.0  4  0.0  −2  −2  0.04  0.1  4 0.0  −2  −1  0.1  0  17  1  2  −2  −1  0  1  2  Figure 1.11: Contour plots: BB2 copula + standard normal margins  0.04  2  0.0  2  4  −4  −2  −2  0  2  4  0.02  2  2  4  0  0.1  2  0.1  0  0.0  6 0.0  6  1 0.  0.04  −4  −2  0.04  −4  −4  0  4  0.08  −2  08  0  1  0.  0.  6  0.02  2  4  4 2  2  0  0.08  0.0  −2  −2  δ = 0.4, θ = 0.6  0.02  0.04  −2  −4  δ = 0.4, θ = 0.4  4  δ = 0.4, θ = 0.1  −4  0.06  0.04  −4  −2  0.04  −4  0  0.1  0.  6  −4  −2  1  0  0  0.  0  0.1  6  0.0  −2  0.08  1 0.  08  0.08  0.02  −4  0.02  2  0.02  2  2  4  δ = 0.1, θ = 0.6  4  δ = 0.1, θ = 0.4  4  δ = 0.1, θ = 0.1  −4  −2  0  2  4  −4  −2  0  2  4  two limits, respectively. Explicit formulas for conditional Gaussian, Student t, Frank, Gumbel, MTCJ and BB2 copulas are presented in what follows. The limits for those conditional copulas are summarized in Table 1.2. For those examples, if 1 ≤ κL < 2, then C1|2 (u|0) =: ςL with 0 < ςL ≤ 1. The limit ςL = 1 for such a κL , except for the conditional Student t copula of which ςL is strictly less than 1. The case ςL = 1 means that the limit of the conditional copula degenerates and becomes a single point at 0 as the conditioning random variable goes to 0. But for the Student t copula that has 0 < ςL < 1, the limit of the conditional copula degenerates to be two points at u = 0 and u = 1, respectively, as the conditioning random variable V = v → 0+ . Looking at Figure 1.7, one may find that the bivariate Student t copula possesses tail dependence in all of the four directions, which explains why the limit of the conditional copula degenerates as two points. In parallel, for upper tails, if 1 ≤ κU < 2, then the limit C1|2 (u|1) degenerates to a single point at u = 1 except that for the Student t copula, it degenerates into two points at u = 0 and u = 1, respectively. If κL = 2, that is, if the copula is lower tail quadrant independent, then the limit 18  of the conditional copula as V = v → 0+ is still a non-degenerate cdf. Similarly, if κU = 2, then the limit is a non-degenerate cdf as well. In the sense of such a limit of conditional copulas, that 1 ≤ κ < 2 indicates a relatively higher degree of positive association in the tail. This is also a reason why we want to use different notions to discriminate intermediate tail dependence (1 < κ < 2) and tail quadrant independence (κ = 2), although in the literature of the extreme value theory, these two cases are both referred to as asymptotic independence or tail independence. Gaussian copula Let (X1 , X2 ) be standard bivariate Normal with correlation . Since X1 |X2 = x ∼ N ormal( x, 1 −  2 ),  the conditional Gaussian copula is  C1|2 (u|v) = Φ  Φ−1 (u) − Φ−1 (v) 1−  2  .  Thus C(u|0) = 1 for 0 < u ≤ 1, and C(u|1) = 0 for 0 ≤ u < 1. Student t copula Let (X1 , X2 ) be standard bivariate Student t distributed with ν the degree of freedom and  the correlation coefficient in the correlation matrix. Since by (5.30) of  McNeil et al. (2005), ν + 1 X2 − x X1 = x ∼ tν+1 (0, 1), ν + x2 1 − 2 the conditional Student t copula is ν+1 Tν−1 (u) − Tν−1 (v) × −1 ν + (Tν (v))2 1− 2  C1|2 (u|v) = Tν+1  Thus C1|2 (u|0) = Tν+1 √ Tν+1 − ν + 1/ 1 −  √ 2  ν + 1/ 1 − for 0 < u < 1.  19  2  .  for 0 < u < 1, and C(u|1) =  Frank copula The conditional Frank copula is straightforward to get, and it is C1|2 (u|v) =  (1 − e−δu )e−δv . 1 − e−δ − (1 − e−δv )(1 − e−δu )  Therefore, C1|2 (u|0) = (1 − e−δu )/(1 − e−δ ) for 0 ≤ u ≤ 1, and C1|2 (u|1) = ((1 − e−δu )e−δ )/[(1 − e−δ )e−δu ] for 0 ≤ u ≤ 1. Gumbel copula The conditional Gumbel copula is 1 C1|2 (u|v) = exp − u ˜δ + v˜δ v  1/δ  1+  u ˜ v˜  δ −1+1/δ  ,  where v˜ := − log v and u ˜ := − log u. Thus, C1|2 (u|0) = 1 for 0 < u ≤ 1, and C1|2 (u|1) = 0 for 0 ≤ u < 1. MTCJ copula The conditional Mardia-Takahasi-Cook-Johnson copula (MTCJ) copula is C1|2 (u|v) = (v −δ + u−δ − 1)−1/δ−1 v −δ−1 , and C1|2 (u|0) = 1 for 0 < u ≤ 1, and C1|2 (u|1) = u1+δ for 0 ≤ u ≤ 1. BB2 copula The conditional BB2 copula is C1|2 (u|v) = 1+δ  −1  log e  δ(v −θ −1)  +e  δ(u−θ −1)  −1  eδ(v  −1/θ−1  e  −θ −1)  δ(v −θ −1)  + eδ(u−θ −1) − 1 −θ )  Thus, C1|2 (u|0) = 1 for 0 < u ≤ 1, and C1|2 (u|1) = u1+θ eδ(1−u 1. 20  v −θ−1  .  for 0 ≤ u ≤  Table 1.2: Tail order and conditional copula Copula Gaussian  κL 2/(1 + )  Student t  1  Frank Gumbel MTCJ BB2  1 1 κU 2/(1 + )  Student t  1  1.6.3  2 1 2 2  √  ν + 1/ 1 −  2  for 0 < u < 1  (1 − e−δu )/(1 − e−δ ) for 0 ≤ u ≤ 1 1 for 0 < u ≤ 1 1 for 0 < u ≤ 1 1 for 0 < u ≤ 1  2 21/δ  Copula Gaussian Frank Gumbel MTCJ BB2  Tν+1  C1|2 (u|0) 1 for 0 < u ≤ 1  Tν+1 − ((1 −  √  C1|2 (u|1) 0 for 0 ≤ u < 1 ν + 1/  e−δu )e−δ )/[(1  −  1−  2  for 0 < u < 1  e−δ )e−δu ]  for 0 ≤ u ≤ 1 0 for 0 ≤ u < 1 u1+δ for 0 ≤ u ≤ 1 −θ u1+θ eδ(1−u ) for 0 ≤ u ≤ 1  Tail order and Conditional Tail Expectation  Two fundamental conditional specifications that are often useful in more specific modelings are E[X1 |X2 > t] and E[X1 |X2 = t]. In actuarial science, these two quantities are often referred to as certain forms of Conditional Tail Expectation, and they are useful as risk measures to assessing magnitude of losses. Moreover, the study of the tail behavior as t → ∞ of these two conditional specifications may also be meaningful when one needs to develop diagnostic plots for discriminating types of tail dependence, or formulate certain regression models that account for tail dependence patterns. The case: E[X1 |X2 > t] Let X1 , X2 be non-negative random variables with copula C and the same Pareto distribution F (x) = 1 − (1 + x)−θ , θ > 1. Now we consider the effect of tail order  21  on the following conditional specification. Letting v := F (t), E[X1 |X2 > t] 1  = (1 − v)−1 1/(θ − 1) +  {[−v + C(u, v)] · [f (F −1 (u))]−1 }du . (1.2)  0  Similar to that we use regularly varying functions to describe the tail behavior of copula functions, we may assume that there exists a regularly varying function g(t) := tβ (t) such that E[X1 |X2 > t] = g(t) = tβ (t),  t → ∞.  Then we can briefly discuss the following three cases: 1. For the usual tail dependence case, β = 1 with some regularity conditions. The derivation of this form can be conducted through the theory of MRV. 2. For the intermediate tail dependence case, with certain regularity conditions, 0 < β < 1 can be a function of the shape parameter α for the Pareto margins. 3. For the tail quadrant independence case, with some regularity conditions, β = 0. A special case is when X1 and X2 are independent, then clearly E[X1 |X2 > t] = E[X1 ] is a constant and does not rely on t. In Figure 1.12, a comparison between Gumbel, s.Gumbel and MTCJ copulas is illustrated. Based on it, we find that when the upper tail order is 1 (Gumbel), the CTE plot seems to be linear in t on the whole support no matter what θ is. When the upper tail order is 2 (MTCJ), the CTE becomes very flat. The CTE plots for the intermediate upper tail dependence with the upper tail order 1 < κ < 2 (s.Gumbel) are located between the above two cases. The case: E[X1 |X2 = t] Let X1 , X2 be non-negative random variables with copula C and the same Pareto distribution F (x) = 1 − (1 + x)−θ , θ > 1. Then we compare the effect of tail order on the following conditional specification. Letting v := F (t), 1  E[X1 |X2 = t] =  1 − C1|2 (u|v) · [f (F −1 (u))]−1 du.  0  22  Figure 1.12: Tail order and CTE plots of the form E[X1 |X2 > t]. The range of t in each plot was chosen to cover the support of t between the 1% and 95% quantiles. Gumbel(δ = 2), s.Gumbel(δ = 2) and MTCJ(δ = 1). Pareto margins with θ = 1.5  Pareto margins with θ = 10  Gumbel s.Gumbel MTCJ  Gumbel s.Gumbel MTCJ  0.4  E( X1 | X2 > t )  E( X1 | X2 > t )  15  10  0.3  0.2 5  0.1 0  2  4  6  0.0  0.1  t  0.2  0.3  t  Figure 1.13: Tail order and CTE plots of the form E[X1 |X2 = t]. The range of t in each plot was chosen to cover the support of t between the 1% and 95% quantiles. Gumbel(δ = 2), s.Gumbel(δ = 2) and MTCJ(δ = 1). Pareto margins with θ = 1.5  Pareto margins with θ = 10 0.30  6  Gumbel s.Gumbel MTCJ  5  4  0.20  E( X1 | X2 = t )  E( X1 | X2 = t )  Gumbel s.Gumbel MTCJ  0.25  3  0.15  2 0.10 1 0.05 0 0  2  4  6  0.0  t  0.1  0.2  t  23  0.3  From Figure 1.13, we can also observe the similar pattern of linearity for the Gumbel copula, and that the line for MTCJ copula becomes flat. Those are just illustrations of the effect of tail order on Conditional Tail Expectation (CTE)s. In Chapter 5, more detailed study will be conducted for comparisons of CTEs under different dependence in the tails; in Chapter 7, more detailed study on such CTE plots are presented to illuminate future research on this topic.  1.6.4  Tail order and tail order functions  In this subsection, we briefly introduce the concepts of tail order and tail order functions for the following three general copula families: extreme value copula, Archimedean copula and elliptical copula. More detailed study is in Chapter 3. Extreme value copula If a copula C satisfies C(ut1 , . . . , utd ) = C t (u1 , . . . , ud ) for any (u1 , . . . , ud ) ∈ [0, 1]d and t > 0, then we refer to C as an extreme value copula. For any extreme value copula C, there exists a function A : [0, ∞)d → [0, ∞) such that C(u1 , . . . , ud ) = exp{−A(− log u1 , . . . , − log ud )}, where A is convex, homogeneous of order 1 and satisfies max(x1 , . . . , xd ) ≤ A(x1 , . . . , xd ) ≤ x1 + · · · + xd . For the bivariate case, C(u, u) = exp{A(1, 1) log u} = uA(1,1) , and the lower tail order is κL = A(1, 1). Also, C(1 − u, 1 − u) = 2u − 1 + (1 − u)A(1,1)−1 ∼ [2 − A(1, 1)]u, u → 0+ . Therefore, the upper order is κU = 1. Tail order functions for extreme value copulas are relatively easy to get. For A (1,1)  w2 2  1 , w2 )  = w1 +  the lower tail, b(w1 , w2 ) = limu→0+ C(uw1 , uw2 )/uA(1,1) = w1 1 where Ai = ∂A(x1 , x2 )/∂xi , i = 1, 2. For the upper tail,  b∗ (w  A (1,1)  ,  w2 − A(w1 , w2 ). Archimedean copula The Archimedean copula studied in this thesis is constructed by a Laplace Transform (LT) ψ of a positive random variable; that is C(u1 , . . . , ud ) = ψ ψ −1 (u1 ) + · · · + ψ −1 (ud ) . 24  This form covers most of the commonly used Archimedean copulas, and it has the following representation ∞ d  Gηj (uj ) dFH (η),  C(u1 , . . . , ud ) = 0  j=1  where FH is the cdf of the resilience random variable H, Gj (u) = exp{−ψ −1 (u)} (0 ≤ u ≤ 1) for all j, and ψ(s) = ψH (s) =  ∞ −sη dFH (η) with lims→∞ ψ(s) 0 e  =  0. When η becomes larger, the density of the joint distribution  η d j=1 Gj (uj )  will  be pushed to the right end of the support; as η goes to a smaller value, the density of the joint distribution  η d j=1 Gj (uj )  will be pushed towards the left end of the  support. Therefore, the behavior of H at ∞ will affect the upper tail of the copula, and the behavior of H at 0 will affect the lower tail of the copula. For a positive random variable Y with LT ψ, if the maximal non-negative moment MY := sup{m ≥ 0 : E(Y m ) < ∞} is non-integer such that 1 < MY < d, then the upper tail order of Cψ is κU = MY under some mild regularity conditions; if 0 ≤ MY < 1, then the upper tail order of Cψ is κU = 1. In other words, the behavior of ψ(u) as u → 0+ influences the upper tail behavior of the copula Cψ . The tail behavior of ψ(u) as u → ∞, however, will affect the lower tail behavior of the copula Cψ . When tail order of an Archimedean copula has been obtained, we can get the corresponding tail order functions usually by applying the l’Hˆopital’s rule. Elliptical copula d  An elliptical random vector X has the stochastic representation X = RAU, where A is a deterministic matrix, and R ≥ 0 is independent of U and referred to as the radial random variable, and U is uniformly distributed on the surface of a unit ball. The tail behavior of R will affect the tail behavior of corresponding elliptical copula. If R is in the Maximum Domain of Attraction (MDA) of Fr´echet, then the copula has tail dependence, i.e., κ = 1; if R is in the MDA of Gumbel, then the tail order κ > 1 and the value of κ relies on both the tail behavior of R and the associated correlation coefficient . 25  Tail order functions of elliptical copulas for the first case (R ∈ MDA(Fr´echet)) can be calculated directly by considering the above stochastic representation, but a general form for the second case remains to be an open question. We refer to Chapter 3 for more detailed arguments about tail order and tail order function for elliptical copulas. Inference for tail probability Tail order functions can be useful for statistical inference on multivariate extremal events. Let b∗ be the upper tail order function for a copula C, and C be the corresponding survival copula, then we have C(uw) ∼ uκ (u)b∗ (w), u → 0+ . Since the term uκ is tractable and dominates the tail behavior, the tail order function b∗ (w) may approximate the tail behavior of the copula C. Tail order functions are limiting functions and it would be more robust than using the copula itself for statistical inference on tail events.  1.6.5  Concordance ordering in the tail  Let X and Y be d-dimensional random vectors with distribution function FX d  and FY such that Xi = Yi , i = 1, . . . , d. Then X is less concordant than Y if FX (z) ≤ FY (z) and F X (z) ≤ F Y (z) for any z in the support of FX and FY . The concordance order is an important concept in the theory of stochastic orders (Joe, 1990). That means small values of Y are more likely to occur together than those of X, and large values of Y are also more likely to occur together than those of X. If X is less concordant than Y, then the upper and lower tail orders of the copula of X are larger than those of the copula of Y, respectively. For comparisons between tails, especially between tails in a single direction, the concordance order is a relatively strong condition. Here we actually only need the conditions such as FX (z) ≤ FY (z) holds ultimately as z is sufficiently small for comparing the lower tail, and F X (z) ≤ F Y (z) holds ultimately as z is sufficiently large for comparing the upper tail.  26  1.6.6  Sub-extremes and second order conditions  In real applications of tail modeling, one often has interests not only in the limiting properties but also in the tail part beyond a high threshold. If the limit is referred to as an extreme, then the tail part beyond the high threshold will be referred to as sub-extremes. We use limiting properties – such as tail index for margins and tail order for dependence structures – as summary quantities for the tail, in hope that the limiting quantity may provide a reasonable approximation for the extreme and sub-extremal levels as well. Apparently, the speed of decay for the convergence involved to get the limiting quantity may influence the goodness of such an approximation. We will use second order conditions as a generic term to represent those conditions that may affect rates of certain convergence. For instance, in Chapter 6, we study second order conditions in the form of the so-called second order regular variation. For example, if g(x) := x−2 + x−3 , then for any given t > 0, g(xt)/g(x) → t−2 as x → ∞. Since g(xt)/g(x) − t−2 = (x + 1)−1 (t−3 − t−2 ), the term (x + 1)−1 ∼ x−1 dominates the speed of convergence of g(xt)/g(x) → t−2 . We refer to −1 here as a second order parameter. When the second order parameter is more negative, the limiting quantity t−2 becomes a better approximation to g(xt)/g(x) for large x.  1.7  Outline  This thesis is organized as following: In Chapter 2, motivations, definitions and related known results for some fundamental concepts and technical tools will be presented. Chapter 3 includes the study of using tail order to quantify the degree of positive tail association. Some relevant studies on multivariate copulas such as the Archimedean copula have also been done in this chapter. In Chapter 4, tail comonotonicity, the strongest tail dependence case will be studied for its fundamental properties, constructions and influence on risk measures. Chapter 5 will be used to report the findings of how tail behavior of copulas affects commonly-used risk measures, and in particular, of the conservativity of tail comonotonicity. In Chapter 6, we will study how 2RV affects the performance of risk measures. At last, in Chapter 7, we will conclude the thesis and propose some relevant topics for future research. 27  1.8  Highlights of selected new results  • A new concept called tail order has been proposed in Definition 3.1, and the corresponding tail order function is defined by Definition 3.2. A formula for deriving tail order of a bivariate elliptical copula where the radial random variable belongs to MDA(Gumbel) is given in Proposition 3.1. Some fundamental properties for these two concepts are given in Proposition 3.2 and 3.3. • In order to study the tail behavior of the Archimedean copula, we first prove a result in Proposition 3.5 that relates the tail heaviness of a positive random variable to the Taylor expansion of its LT. • For the Archimedean copula, we relate the tail heaviness of a positive random variable to the tail behavior of the Archimedean copula constructed from the LT of the random variable, and extend the results of Charpentier and Segers (2009) for the upper tail in Proposition 3.6. And the corresponding upper tail order function has been proved in Proposition 3.7 as well. For the lower tail, we study a more concrete form by assuming some mild conditions, and the relevant result is Proposition 3.8. • A new notion of tail comonotonicity is proposed in Definition 4.1, and relevant results about their construction include Proposition 4.3 and Proposition 4.8. • Asymptotic additivity of risk measures Value at Risk (VaR) and CTE under tail comonotonicity is reported in Propositions 4.13, 4.14, 4.15 and 4.16. • Effect of tail behavior of copulas on E[X1 |X2 > t] and E[X1 |X2 = t] are mainly summarized in Propositions 5.1, 5.4 and 5.9, and their corollaries. • For the study of 2RV, results for asymptotic analysis of CTE and VaR has been reported in Proposition 6.4 for the univariate case, and in Proposition 6.6 for the multivariate case.  28  Chapter 2  Preliminaries 2.1  Risk measures  Some statistical quantities about a random variable X are often referred to as risk measures. Among many of them, Value at Risk (VaR) and conditional tail expectation (CTE) are probably the most popular risk measures. Both of them have been adopted by regulations of insurers. For example, CTE has been required for calculating the relevant risks of segregated fund in Canada (OSFI, 2011). Definition 2.1 Let be given a random variable X the amount of risk, and p ∈ (0, 1) a probability level, then the corresponding VaR, denoted by VaRp (X), is defined as VaRp (X) = FX−1 (p) := inf{x ∈ R : FX (x) ≥ p}. If E[X] < ∞, then the corresponding CTE is defined as CTEp (X) = E[X|X > VaRp (X)].  (2.1)  Note that the VaR defined above is a left-continuous generalized inverse of FX , and CTE is a coherent risk measure for continuous random variables (Artzner et al., 1999), and it can better account for tail heaviness of a loss distribution than VaR does. We refer to Denuit et al. (2005) for some other relevant risk measures and their relationships. 29  Risk measures are used to help people make decisions through certain decision principles. For instance, one use risk measures to determine premiums of insurance products or calculate solvency capital charges. We refer to Goovaerts et al. (2010) for a comprehensive overview of risk measures and relevant decision principles. The risk measures mentioned in Definition 2.1 are both quantile-based risk measures. In this thesis, the emphasis will be risk measures for high-risk scenarios, for which p is close to 1. We will conduct asymptotic comparisons of risk measures in the sense that, when p is close to 1 what will be the ordering for those risk measures. For stochastic comparisons of risks, M¨uller and Stoyan (2002) is popular reference. Recently, Mainik and R¨uschendorf (2012) conduct asymptotic comparisons for risks of portfolios, where a concept of asymptotic portfolio loss order is studied. A classic question in quantitative risk management is to study the so-called diversification benefit of a portfolio of risks. Definition 2.2 Let X := (X1 , . . . , Xd )T be a d-dimensional random vector. Then risk concentration  (X|Qp ) of X with respect to a risk measure Qp at level p ∈  (0, 1) is defined as (X|Qp ) :=  Qp (X1 + · · · + Xd ) d i=1 Qp (Xi )  ,  where Qp : R → R+ is a risk measure associated with a probability level p. The corresponding diversification benefit is 1 −  (X|Qp ).  Here two relevant concepts are subadditivity and superadditivity of risk measures. If  (X|Qp ) ≤ 1, then Qp is said to be subadditive with respect to X, and  (X|Qp ) ≥ 1 superadditive. It is well known that CTE defined in (2.1) is subadditive with respect to continuous margins, but VaR is not necessarily subadditive for such a general case (e.g., Denuit et al., 2005). In this thesis, we will consider asymptotic additivity of CTE and VaR in the sense that p → 1− in the risk concentration  (X|Qp ).  30  2.2  Copula  The study of tail behavior of random vectors has received increasing attention, especially in the framework of quantitative risk management. Let X = (X1 , . . . , Xd )T be a random vector with distribution function F and continuous univariate marginal distribution functions Fi , i = 1, . . . , d. Due to Sklar’s theorem Sklar (1959), F (x1 , . . . , xd ) = C(F1 (x1 ), . . . , Fd (xd )),  (2.2)  in which the copula function C : [0, 1]d → [0, 1] is uniquely determined by C(u1 , . . . , ud ) = F (F1−1 (u1 ), . . . , Fd−1 (ud )),  (2.3)  where Fi−1 is the inverse function of Fi , i = 1, . . . , d. The corresponding survival function C is defined as C(u1 , . . . , ud ) = 1 +  |I| ∅=I⊂{1,...,d} (−1) CI (ui , i  ∈ I),  where CI is the I-margin of the copula C with |I| the cardinality of the set I. There exist several related methodologies. The lower tail dependence parameter is defined as λL = lim u−1 C(u, . . . , u), u→0+  and the upper tail dependence parameter is defined similarly with the survival function C. As extensions, Juri and W¨uthrich (2002, 2003) studied tail dependence from a distributional point of view. Kl¨uppelberg et al. (2008) defined the so-called tail dependence function of X as λX (x1 , . . . , xd ) = lim t−1 P[1 − F1 (X1 ) ≤ tx1 , . . . , 1 − Fd (Xd ) ≤ txd ], t→0  and Nikoloulopoulos et al. (2009); Joe et al. (2010); Li and Sun (2009) further studied the properties of the tail dependence function and their applications for multivariate t copulas, vine copulas and heavy-tailed scale mixtures of multivariate distributions, respectively. We refer to the above papers for details and properties of tail dependence functions. Multivariate Archimedean copulas are widely used in insurance and financial risk analyses as they can be used for sensitivity analyses to changes in depen31  dence, tail asymmetry, and lower/upper tail behavior. They are not flexible enough as models for high-dimensional data but can be extended to more flexible copula families (Joe and Hu, 1996) based on mixtures of max-infinitely divisible distributions. In the literature, a d-dimensional Archimedean copula C is often (e.g., Genest and MacKay, 1986; Nelsen, 2006) defined as C(u1 , . . . , ud ) = φ−1 (φ(u1 ) + · · · + φ(ud )), (u1 , . . . , ud ) ∈ [0, 1]d . McNeil and Neˇslehov´a (2009) showed that d-monotonicity is a sufficient and necessary condition on the Archimedean generator φ−1 so that the above form is a copula; a real-valued function g defined on (a, b) with a, b ∈ R is said to be d-monotone (d ≥ 2) if it is differentiable up to the order d − 2, (−1)k g (k) (x) ≥ 0 for any x ∈ (a, b) and k = 0, . . . , d − 2, and (−1)d−2 g (d−2) is non-increasing and convex in (a, b). To get a better understanding of tail dependence (intermediate or usual tail dependence), we use the mixture of power or LT representation in Marshall and Olkin (1988) and Joe (1997); for mixing distribution functions [resp. survival functions], the power is called resilience [resp. frailty] in Marshall and Olkin (2007). Let Cψ (u1 , . . . , ud ) = ψ(ψ −1 (u1 ) + · · · + ψ −1 (ud )),  (u1 , . . . , ud ) ∈ [0, 1]d , (2.4)  where ψ is the LT of a positive random variable. Note that as d ≥ 3 increases, Archimedean copulas extend less into the region of negative dependence (Joe, 1997, Sections 4.4 and 5.4) and hence the restriction to LTs does not lose much generality for defining an Archimedean copula. Since a LT is completely monotone, it can be used to construct copulas of any dimension. In this thesis, we assume that the function ψ is a LT of a positive random variable; that is, ψ(t) :=  ∞ −xt FX (dx), 0 e  where FX is the distribution function  of a positive random variable X. Such choices of ψ cover most of the commonly used Archimedean copulas. The tail behavior of a LT function plays a critical role in analyzing the dependence in the tails of the Archimedean copula constructed by the LT function. Here we list basic properties of a LT ψ : [0, ∞) → [1, 0) as follows: ψ(0) = 1, lims→∞ ψ(s) = 0, ψ is continuous and strictly decreasing, and ψ(s) is completely monotone; that is, the derivatives ψ (n) exist for n = 0, 1, 2, . . . and  32  (−1)n ψ (n) (s) ≥ 0 for all s ≥ 0. If C(u1 , . . . , ud ) is a copula, then the corresponding survival copula is defined as (−1)|I| CI (1 − ui , i ∈ I),  C(u1 , . . . , ud ) = 1 +  I⊂{1,...,d}  where CI is the I-margin of the copula C with |I| the cardinality of the set I; in particular, for bivariate case, C(u, v) = u + v − 1 + C(1 − u, 1 − v). If Cψ is an Archimedean copula constructed by LT ψ, then the corresponding survival copula is referred to as a survival Archimedean copula constructed by LT ψ. The survival copula itself is a copula, and it should be distinguished from the survival function of a copula (denoted as C); for bivariate case, the survival function of a copula C is C(u, v) = 1 − u − v + C(u, v). Definition 2.3 A copula C is said to be reflection symmetric if the copula is the same as its survival copula C; that is, C(u) ≡ C(u) for any u ∈ [0, 1]d . Another copula family that will be studied is extreme value copula. If a copula C satisfies C(ut1 , . . . , utd ) = C t (u1 , . . . , ud ) for any (u1 , . . . , ud ) ∈ [0, 1]d and t > 0, then we refer to C as an extreme value copula. When we compare certain properties for different copulas, we usually assume that those copulas share some common summary quantities such as the following Blomqvist’s β (Blomqvist, 1950) for the bivariate case. Definition 2.4 The Blomqvist’s β for a bivariate copula C is defined as β := 4C(1/2, 1/2) − 1. So −1 ≤ β ≤ 1; β = 0 holds for the independent copula and a larger positive β suggests a higher positive dependence. Moreover, β(C) = β(C) for the bivariate case.  33  Standard references for copula theory are Joe (1997) and Nelsen (2006). For applications of copulas in actuarial science and quantitative risk management, we refer to Denuit et al. (2005), McNeil et al. (2005) and Genest et al. (2009a).  2.3  Regular variation  For asymptotic analysis of tail behavior of random variables or equivalently their distribution functions, the theory of regular variation provides a powerful platform. Here we only give some most fundamental concepts. More relevant concepts and results about regular variation will be used throughout the thesis, and will be introduced when they are needed. Standard references on regular variation are Bingham et al. (1987), Resnick (1987), and Geluk and de Haan (1987); Embrechts et al. (1997) and Resnick (2007) are more relevant for applications in actuarial science, quantitative finance and risk management. Definition 2.5 A measurable function g : R+ → R+ is regularly varying at ∞ with index α = 0 (written g ∈ RVα ) if for any t > 0, lim  x→∞  g(xt) = tα . g(x)  (2.5)  If equation (2.5) holds with α = 0 for any t > 0, then g is said to be slowly varying at ∞ and written as g ∈ RV0 . For the lower limit at 0+ , if for any t > 0, limx→0+ g(xt)/g(x) = tα , then g is regularly varying at 0+ and denoted by g ∈ RVα (0+ ). Note that g(t) ∈ RVα ⇐⇒ g(1/t) ∈ RV−α (0+ ). Similarly, RV0 (0+ ) is defined. We will use (x) to represent a slowly varying function, and a regularly varying function g can be written as g(x) = xα (x). Note that, when we say that a random variable is regularly varying, it actually means that the survival function of the random variable is regularly varying. Definition 2.6 A measurable function g : R+ → R+ is rapidly varying at ∞ with  34  index ∞ (written g ∈ RV∞ ) if for any t > 0, g(xt) lim = t∞ x→∞ g(x)     0, = 1,   ∞,  if t < 1, if t = 1, if t > 1.  Similarly, g ∈ RV−∞ if for any t > 0, g(xt) lim = t−∞ x→∞ g(x)     ∞, = 1,   0,  if t < 1, if t = 1, if t > 1.  Proposition 2.1 If U ∈ RVα for α ∈ R, then lim U (tx)/U (x) = tα ,  x→∞  locally uniformly in t on (0, ∞). If α < 0, then uniform convergence holds on intervals of the form (b, ∞), b > 0. If α > 0, uniform convergence holds on intervals (0, b] provided U is bounded on (0, b] for all b > 0. The following result is the famous Karamata’s theorem. It tells us that if we write some regularly varying function U (x) = xα (x) with (x) slowly varying, then U (x) behaves like xα in terms of integration. Theorem 2.2 Suppose α ≥ −1 and U ∈ RVα . Then lim  x→∞  If α < −1 (or if α = −1 and ∞ x U (t)dt  is finite,  ∞ x U (t)dt  lim  x→∞  xU (x)  x 0 U (t)dt  = α + 1.  x 0 U (t)dt ∞ x U (s)ds  ∈ RVα+1 and (2.6)  < ∞), then U ∈ RVα implies that  ∈ RVα+1 , and xU (x) ∞ x U (t)dt  = −α − 1.  (2.7)  Definition 2.7 A function g : R → (0, ∞) is Γ-varying if it is non-decreasing and right-continuous, and there exists a measurable function h : R → (0, ∞) such that 35  for any t ∈ R g(x + th(x)) = et , x→∞ g(x) lim  (2.8)  where h(·) is called an auxiliary function. Definition 2.8 A measurable function g : R+ → R is Π-varying if there exits a function h : R+ → R+ such that for all t > 0, lim  x→∞  g(tx) − g(x) = log t, h(x)  (2.9)  where h(·) is called an auxiliary function. The following is a definition for MRV. Due to the limitation of space, we will not give more details about MRV here. For interested readers, Resnick (2007) is a nice reference. Definition 2.9 A random vector X ≥ 0 is MRV if there exists a Radon measure µ (i.e., finite on compact sets) on E := [0, ∞] \ {0} such that, lim  t→∞  P[X ∈ tB] = µ(B) P[||X|| > t]  for any relatively compact set B ⊂ E with µ(∂B) = 0, where || · || denotes a norm on Rd .  2.4  Maximum Domain of Attraction  Definition 2.10 A random variable X is said to belong to the MDA of an extreme value distribution H if there exist normalizing constants σn > 0 and µn ∈ R such that d  (Mn − µn )/σn → H,  n → ∞,  where Mn := X[1] is the first order statistics (i.e., maximum) of a random sample of X with the sample size being n. This is written as X ∈ MDA(H). 36  It is well known that there are only three non-degenerate univariate extreme value distributions: Fr´echet, Gumbel and Weibull. Since MDA(Weibull) corresponds to bounded random variables that are irrelevant to quantitative risk management, we will only consider the first two cases. The following two theorems characterize the classes of MDA(Fr´echet) and MDA(Gumbel), respectively. Theorem 2.3 A random variable X with cdf F is said to belong to the Fr´echet maximum domain of attraction if and only if F ∈ RV−α , α > 0, and the corresponding Fr´echet cdf is exp(−x−α ). Theorem 2.4 A random variable X with cdf F is said to belong to the Gumbel maximum domain of attraction if and only if there exists a positive auxiliary function a(·) such that lim  x→∞  F (x + ta(x)) = e−t , F (x)  where a(·) can be chosen as a(x) =  t ∈ R,  (2.10)  ∞ x F (t)/F (x)dt.  The concept of MDA(Gumbel) here is closely related to Γ-variation: if g(x) := 1/F (x) with F satisfying (2.10), then g(x) is non-decreasing and right-continuous, and satisfies the Γ-variation condition (2.8) with h(·) = a(·) the auxiliary function in (2.10). Note that, Marshall and Olkin (1983) has a characterization of MDA of a multivariate extreme value distribution. This paper also shows that a random vector X that follows a multivariate extreme value distribution must be associated; that is, for any real-valued increasing functions g1 , g2 , E[g1 (X)g2 (X)] ≥ E[g1 (X)]E[g2 (X)], provided that the expectations exist.  37  Chapter 3  Tail order and intermediate tail dependence 3.1  Introduction  For statistical modeling with copulas, properties such as strengths of upper/lower tail dependence and reflection symmetry or direction of reflection asymmetry are important in deciding on appropriate copulas. For example, for the tail asymmetry phenomena of financial markets (Patton, 2006; Okimoto, 2008), copula families with a variety of tail behavior are useful for statistical modeling. Although the multivariate Gaussian and t copula families have a wide range of dependence, they are not appropriate when there is reflection or tail asymmetry. But copulas can be constructed from other methods to get different joint tail behavior. Then for use of copulas for inference for joint tail probabilities, sensitivity analysis over different families can be performed. For the study of tail dependence behavior of random vectors, we not only have interest in the cases where the random vector is asymptotically dependent, but also where asymptotic independence exhibits. Ledford and Tawn (1996) proposed the following model for a bivariate random vector (X1 , X2 )T , where X1 and X2 are unit Fr´echet distributed with cdf Fi (x) = e−1/x , x ≥ 0, i = 1, 2, and are non-  38  negatively associated, P[X1 > r, X2 > r] ∼ (r)r−1/η ,  r → ∞,  (3.1)  where 1/2 ≤ η ≤ 1. If we let Ui = Fi (Xi ), i = 1, 2, where Fi is the cdf of the unit Fr´echet and r = (− log(u))−1 , then clearly lim  u→1−  P[U1 > u, U2 > u] P[X1 > r, X2 > r] = lim κ r→∞ (P[U1 > u]) (P[X1 > r])κ  (3.2)  (r)r−1/η (r)r−1/η = lim . r→∞ [1 − exp(−r −1 )]κ r→∞ r−κ  = lim  Thus the “tail order” κ that we will introduce in Definition 3.1 corresponds to 1/η of Ledford and Tawn’s representation. If η = 1, i.e., κ = 1 and (r)  0,  X1 and X2 are upper tail dependent with upper tail dependence parameter λU = limr→∞ (r); if 1/2 < η < 1, they are positively dependent; if η = 1/2 and (r) ≥ 1, they are “tail quadrant independence”. A lot of research has been done following this direction. We refer to Ledford and Tawn (1996, 1997); Coles et al. (1999); Heffernan (2000); Ramos and Ledford (2009) for further development of this idea. The relation (3.1) tells us that the power term 1/η dominates the speed of decay of the joint tail probability. We believe that the parameter 1/η plays an important role in the study of tail dependence behavior, and deserves a new name “tail order” that is explained in Section 3.2.1, based on copula functions. Moreover, analogously to the tail dependence function, we will propose the tail order function, which includes the information of the convergence along routes other than the diagonal. In this chapter, the emphasis is on the case where the tail order is between 1 and d for a d-dimensional random vector. We refer to this case as “intermediate tail dependence” under some positive dependence assumptions; this is explained before Example 3.1. Our main contributions span the following aspects: 1. We propose the concepts of tail order and tail order functions as an integrated way to study tail behavior of multivariate copulas. 2. We relate the tail heaviness of a positive random variable  39  to the tail behavior of the Archimedean copula constructed by the LT of the random variable. In our opinion, it is an insightful way to better understand the tail behavior of Archimedean copulas. 3. Our theoretical study of tail behavior of Archimedean copulas leads to a new one-parameter Archimedean copula family, based on the LT of the inverse Gamma distribution, which shows patterns of upper and lower tails not seen in commonly used copula families. The remainder of this chapter is organized as follows. Section 3.2 introduces the concepts of tail order and tail order functions, and some properties of them. In particular, some results on relations of tail orders of marginal copulas are given. Sections 3.3 and 3.4 contain studies of intermediate tail dependence for Archimedean copulas and copulas constructed by mixture of max-id distributions, respectively. For multivariate Archimedean copulas, we have a more concrete result than Charpentier and Segers (2009) for the lower tail, and new results for the upper tail. Asymptotic behavior of LTs of positive random variables is studied in Section 3.3.1, and the new Archimedean copula family is presented in Section 3.3.4. Finally, Section 3.5 concludes with some topics of further research. The main proofs are put in Section 3.6.  3.2  Tail orders: definitions and properties  In this section, we define the concepts of tail order and tail order functions, indicate their use for reflection asymmetry and derive some properties.  3.2.1  Multivariate tail order and tail order functions  To avoid technicalities for tail orders, we assume conditions involving regular variation of tails of copula and other functions. Definition 3.1 Suppose C is a d-dimensional copula. If there exists some κL (C) > 0 such that, with some (u) ∈ RV0 (0+ ) C(u1d ) ∼ uκL (C) (u),  u → 0+ ,  then we refer to κL (C) as the lower tail order of C and refer to λL (C) = limu→0+ (u) as the lower tail order parameter, provided the limit exists. Similarly, the upper 40  tail order is defined as κU (C) such that C((1 − u)1d ) ∼ uκU (C) (u),  u → 0+ ,  with the upper tail order parameter λU (C) = limu→0+ (u), provided the limit exists. When no confusion arises, we use the notation κ to represent lower or upper tail orders, and λ for tail order parameters. κL (C) = 1 [resp. κU (C) = 1] and (u) 0 corresponds to the usual definition of upper [resp. lower] tail dependence. We will assume that lims→0+ (s) = h ∈ [0, ∞]. But h = 0 or h = ∞ correspond to boundary cases, in which case more care is needed. In these boundary cases, the “speed” of decrease or increase of (u) affects the tail dependence behavior. For example, if (u) → 0, then a lower speed indicates a stronger tail dependence; if (u) → +∞, then a higher speed indicates a stronger tail dependence. Note that with (u) → h, if κ(C) = 1 then 0 ≤ h ≤ 1; if κ(C) > 1 then 0 ≤ h ≤ ∞. Note also that κL (C) = κU (C) = d for the d-dimensional independence copula. It is not possible for κ < 1 (refer to Proposition 3.3), but it is possible for κL (C) and κU (C) to be greater than d for copulas with negative dependence. For example, as a boundary case, for the bivariate counter-monotonic copula, κL (C) and κU (C) can be considered as +∞ because C(u, u) and C(1 − u, 1 − u) are zero for 0 < u < 1/2. The cases of κ = 1 or d have been well studied in the literature, while not much research exists for 1 < κ < d. For the bivariate case, 1 < κ < 2 represents some level of positive dependence in the tail, but not as strong as tail dependence. For multivariate cases, without any further conditions, the meaning of 1 < κ < d is complicated. We refer to the case 1 < κ < d as lower [resp. upper] intermediate tail dependence only when all marginal copulas (ultimately) possess positive lower [resp. upper] orthant dependence, of which a formal definition will be given in Definition 3.3. Unless otherwise specified, when a copula is said to possess intermediate tail dependence, the orthant dependence condition is assumed implicitly. The following is an example of intermediate tail dependence for Gaussian copulas.  41  Example 3.1 (Gaussian copula) Consider a multivariate Gaussian copula, constructed by CΦd (u1 , . . . , ud ) = Φd (Φ−1 (u1 ), . . . , Φ−1 (ud ); Σ),  (3.3)  where Φd (·; Σ) is the joint cdf of a standard d-variate Gaussian random vector with positive definite correlation matrix Σ. The multivariate Gaussian copula defined in (3.3) has intermediate tail dependence with the tail order κ = 1d Σ−1 1T d , the sum of all elements of Σ−1 . It can be verified by noticing that limu→0+ CΦd (u1d )/uκ = limt→−∞ Φd (t1d )/[Φ(t)]κ , and as t → −∞, Φ(t) ∼ φ(t)/|t| and Φd (t1d ) is domusler, 2003, inated by the exponent term exp(−t2 1d Σ−1 1T d /2) (Hashorva and H¨ Corollary 4.1). The bivariate Gaussian copula with  > 0 has intermediate tail dependence  with the tail order κ = 2/(1 + ) and the slowly varying function at 0+ being (u) = (− log u)−  /(1+ ) .  A related result without using copula functions has  been given in Ledford and Tawn (1996). For dimension d with constant correlation , the tail order is κ = d/[1 + (d − 1) ]. For the trivariate case with we have κ = [3 +  13  − 4 ]/[1 +  13  −2  12  =  23  = ,  2 ].  Gaussian copulas are reflection symmetric and have intermediate tail dependence when correlations are positive. They are a subfamily of the elliptical copulas. Under some regularity conditions, tail orders of elliptical copulas will be determined by the tail behavior of corresponding radial random variable R. Since copula is invariant to a strict increasing transformation on margins, for the study of elliptical copula, we may omit the location and scale parameters of joint elliptical distributions, and consider the following representation: let X := (X1 , X2 ) be an elliptical random vector such that d  X = RAU,  (3.4)  where the radial random variable R ≥ 0 is independent of U, U is an bivariate random vector uniformly distributed on the surface of the unit hypersphere {z ∈ Rk |zT z = 1}, A is a 2 × 2 matrix such that AAT = Σ where Σ11 = Σ22 = 1  42  and Σ12 = Σ21 =  with −1 <  < 1, eg., A =  1  0  . For such an 1− 2 elliptical distribution, the margins have the same distribution assumed to be F . The tail behavior of R may influence the tail order of elliptical copulas. For the usual tail dependence case, Schmidt (2002) proved that when the radial random variable R has a regularly varying tail then X1 and X2 are tail dependent, and thus the tail order of the corresponding elliptical copula is κ = 1. Example 3.2 (Student t copula) The radial random variable R for Student t distributions is a generalized inverse Gamma distribution such that R2 follows an inverse Gamma distribution with the shape and scale parameters being ν/2, where ν is the degree of freedom. It can be verified that F R ∈ RV−ν (see Example 6.4). So the tail order for Student t copula is κ = 1. For the case where R has lighter tails than any regularly varying tails, some asymptotic study has been conducted for elliptical distributions where R belongs to the Gumbel Maximum Domain of Attraction. We refer to Hashorva (2007), Hashorva (2010) and Hashorva (2008) for relevant references. Here we are ready to present a result that is useful to find the tail order of a bivariate elliptical copula where the radial random variable R belongs to Gumbel MDA. Proposition 3.1 Let C be the copula for an elliptical random vector X := (X1 , X2 ) 2/(1 + ). If R ∈ MDA(Gumbel), then the up-  constructed as (3.4), and b = per and lower tail orders of C is  κ = lim  r→∞  log (1 − FR (b r)) , log (1 − FR (r))  (3.5)  provided the limit exists. This result is very convenient for us to derive the tail order if we know the tail behavior of R. Example 3.3 (Bivariate symmetric Kotz type (Fang et al., 1990) copula) The den-  43  sity generator g(x) = KxN −1 exp{−βxξ },  β, ξ, N > 0,  where K is a normalizing constant. By Theorem 2.9 of Fang et al. (1990), the density function of R is fR (x) = 2πxg(x2 ) = 2Kπx2N −1 exp{−βx2ξ }. So, the survival function is ∞  1 − FR (x) = x  2Kπt2N −1 exp{−βt2ξ }dt  ∞  = βx2ξ  Kπ −N/ξ N/ξ−1 β w exp{−w}dw ξ  Kπ −N/ξ β Γ(N/ξ, βx2ξ ), Γ(·, ·) incomplete Gamma function ξ Kπ −1 2N −2ξ ∼ β x exp{−βx2ξ }, x → ∞. ξ =  Then by (3.5), we can easily get that κ = b2ξ = [2/(1 + )]ξ . Therefore, the tail order for the symmetric Kotz type copula is κ = [2/(1 + )]ξ . Gaussian copula belongs to this class with ξ = 1, so its tail order is 2/(1 + ) which is consistent to what we have obtained. Definition 3.2 Suppose C is a d-dimensional copula and C(u1d ) ∼ uκ (u), u → 0+ for some (u) ∈ RV0 (0+ ). The lower tail order function b : Rd+ → R+ is defined as b(w; C, κ) = lim  u→0+  C(uwj , 1 ≤ j ≤ d) , uκ (u)  provided the limit function exists. In parallel, if C((1 − u)1d ) ∼ uκ (u), u → 0+ for some (u) ∈ RV0 (0+ ), the upper tail order function b∗ : Rd+ → R+ is defined  44  as b∗ (w; C, κ) = lim  u→0+  C(1 − uwj , 1 ≤ j ≤ d) , uκ (u)  provided the limit function exists. If (u) → h = 0, then hb(w; C, 1) and hb∗ (w; C, 1) become the tail dependence functions in Joe et al. (2010). Note that the copula C that satisfies the conditions of the above definition is said to be multivariate regularly varying with a limit function b or b∗ (Resnick, 2007). Although the general theory and definitions must accommodate an arbitrary slowly varying function , in specific parametric families of copulas that have tractable forms, we find that either (u) is a constant or proportional to a power of (− log u). Example 3.4 (Extreme value copula) For any multivariate extreme value copula C EV , there exists a function A : [0, ∞)d → [0, ∞) such that C EV (u1 , . . . , ud ) = exp{−A(− log u1 , . . . , − log ud )}, where A is convex, homogeneous of order 1 and satisfies max(x1 , . . . , xd ) ≤ A(x1 , . . . , xd ) ≤ x1 + · · · + xd . We refer to Chapter 6 of Joe (1997) for details of multivariate extreme value copulas. Thus, C EV (u1d ) = exp{A(1d ) log u} = uA(1d ) . That is, for any extreme value copula C EV , the lower tail order is κL (C EV ) = A(1d ) and there is intermediate lower tail dependence except for the boundary cases such as independence copula and comonotonicity copula, where A(1d ) = d and 1, respectively. In order to get the lower tail order function of extreme value copulas, first  45  consider the bivariate case, for which C EV (uw1 , uw2 ) = exp{−A(− log uw1 , − log uw2 )} log w1 log w2 ,1 + log u log u log w1 ∼ exp (log u) A(1, 1) + A1 (1, 1) log u = exp (log u)A 1 +  A (1,1)  = uA(1,1) w1 1  A (1,1)  w2 2  + A2 (1, 1)  log w2 log u  , u → 0+  ,  where Ai = ∂A/∂xi , i = 1, 2. Therefore, the lower tail order function is b(w1 , w2 ) = A (1,1)  w1 1  A (1,1)  w2 2  . Similarly, for a d-variate extreme value copula, b(w1 , . . . , wd ) A1 (1d ) Ad (1d ) w1 . . . wd . By Euler’s formula for homogeneous functions, A(1d ) d i=1 Ai (1d ). Then it can be verified that b is homogeneous of order A(1d ).  = =  In the bivariate case, κU = 1, λU = 2 − A(1, 1), and κL = A(1, 1). That is, a larger value of the upper tail dependence parameter implies stronger lower intermediate tail dependence. Example 3.5 (Elliptical copula) Tail order functions of bivariate elliptical copulas can be derived as following: for any given 0 < w1 , w2 < ∞, as 0 < v < 1 is sufficiently close to 0, C(1 − vw1 , 1 − vw2 ) = P[F (X1 ) ≥ 1 − vw1 , F (X2 ) ≥ 1 − vw2 ] = P[X1 ≥ F −1 (1 − vw1 ), X2 ≥ F −1 (1 − vw2 )] = P[R cos ϕ ≥ F −1 (1 − vw1 ), R[ cos ϕ + =  1 2π  P R≥ θ∈Θ1  F −1 (1 − vw1 ) ,R ≥ cos θ  1−  2 sin ϕ]  ≥ F −1 (1 − vw2 )]  F −1 (1 − vw2 ) cos θ +  1−  2 sin θ  dθ,  where ϕ ∼ Uniform(−π, π) and Θ1 := (−π/2, π/2)∩{θ : tan θ ≥ − /  46  1−  2 }.  Also, letting Θ2 := (−π/2, π/2), as 0 < v < 1 is sufficiently close to 0 v = P R cos ϕ ≥ F −1 (1 − v) =  1 2π  P R≥ θ∈Θ2  F −1 (1 − v) dθ. cos θ  If R ∈ MDA(Fr´echet), i.e., R ∈ RV−α for some α > 0, then the margins X1 , X2 ∈ RV−α . Therefore, the map v → F −1 (1 − v) ∈ RV−1/α (0+ ). Then, by the uniform convergence of regularly varying functions (e.g., Proposition 2.4 of Resnick (2007)) lim  v→0+  C(1 − vw1 , 1 − vw2 ) v θ∈Θ1  P R≥  F −1 (1−vw1 ) cos θ  = lim  v→0+ θ∈Θ2  P R≥  ∨  F −1 (1−vw2 ) √ cos θ+ 1− 2 sin θ  F −1 (1−v) cos θ  w1  F −1 (1−v) cos θ  = lim  v→0+ θ∈Θ2  =  θ∈Θ1  P R≥  dθ/P [R > F −1 (1 − v)] −1/α  −1/α  θ∈Θ1 P R ≥  ∨  w2  α θ∈Θ2 (cos θ) dθ  −1 (1−v)  F √  cos θ+  F −1 (1−v) cos θ  max w1 (cos θ)α , w2 ( cos θ +  dθ/P R > F −1 (1 − v)  1−  2  sin θ  dθ/P R > F −1 (1 − v)  dθ/P [R > F −1 (1 − v)]  1−  2 sin θ)α  dθ .  The multivariate case is similar (Kl¨uppelberg et al., 2008). However, if R ∈ MDA(Gumbel), a general form of tail order functions of the whole class remains unsolved. We next mention how the upper and lower tail orders are useful to establish the direction of reflection asymmetry. Let C be the copula of (1 − U1 , . . . , 1 − Ud ) when the copula of (U1 , . . . , Ud ) is C, where Ui ’s are standard uniform variables. Reflection symmetry means that C ≡ C (Definition 2.3) and otherwise we say that there is reflection asymmetry. If C(u1d ) ≥ C(u1d ) for all 0 < u < u0 , for some 0 < u0 ≤ 1/2, then the copula has more probability in the lower tail (reflection asymmetry with skewness to lower tail). If the inequality is reversed leading to C(u1d ) ≤ C(u1d ), then the copula has more probability in the upper tail (reflection asymmetry with skewness to upper tail). For most existing parametric families  47  of copulas, it is difficult to analytically compare C(u1d ) and C(u1d ), so the direction of reflection asymmetry is analytically easier via the upper and lower tail orders. For example, if κL (C) > κU (C), then C has reflection asymmetry skewed to the upper tail (smaller κ means slower convergence to 0), and if C(u1d ) ∼ λL uκ and C(u1d ) ∼ λU uκ as u → 0+ with λL > λU > 0, then C has reflection asymmetry skewed to the lower tail. For many parametric copula families where we have done numerical computations, u0 can be taken as 1/2. The following are some elementary properties of the lower and upper tail order functions b and b∗ . Obvious properties of tail order for C are the following: κL (C) = κU (C), κU (C) = κL (C), b(w; C, κ) = b∗ (w; C, κ) and b∗ (w; C, κ) = b(w; C, κ). Proposition 3.2 A lower tail order function b(w) = b(w; C, κ) has following properties: 1. b(1) ≡ 1, and b(w) = 0 if there exists an i ∈ Id with wi = 0; 2. b(w) is increasing in wi , i ∈ Id ; 3. for any fixed t > 0, C(tuwj , 1 ≤ j ≤ d) uκ (u) C(tuwj , 1 ≤ j ≤ d) = tκ lim = tκ b(w). (tu)κ (tu) u→0+  b(tw) = lim  u→0+  (3.6)  Thus, b(w) is homogeneous of order κ. If b(w) is partially differentiable with respect to each wi on (0, +∞), then by the Euler’s formula on homogeneous functions, we can write b(w) =  1 κ  d j=1  ∂b wj , ∂wj  ∀w ∈ Rd+ .  Remark 3.1 Since C(uw) ∼ uκ (u)b(w) = b(uw) (u), u → 0+ , the tail order function b captures the tail behavior of the copula C in different directions. 48  3.2.2  Further properties of tail orders  In this subsection, we obtain some general properties of tail orders of multivariate copulas, especially on inequalities on tail orders of marginal copulas. There is an “obvious” property in terms of concordance. For two joint cdfs H, G ∈ R(F1 , . . . , Fd ), we say that H is less concordant than G, if H(x) ≤ G(x) and H(x) ≤ G(x) for any x in the support of H and G. If C1 is less concordant than C2 , then κL (C1 ) ≥ κL (C2 ) and κU (C1 ) ≥ κU (C2 ). Next we introduce some concepts of positive dependence, under which multivariate copulas may have some particular properties on tail orders. We refer to Joe (1997), Colangelo et al. (2005) for details. Definition 3.3 Suppose that F (x) is the cdf of a d-variate random vector X = (X1 , . . . , Xd )T , then X or F is said to be 1. positive lower orthant dependent (PLOD) if P[Xi ≤ xi , ∀i ∈ Id ] ≥ xi ] for any x ∈  d i=1 P[Xi  Rd ;  2. left tail decreasing in sequence (LTDS) if P[Xi ≤ xi |X1 ≤ x1 , . . . , Xi−1 ≤ xi−1 ] is decreasing in x1 , . . . , xi−1 for all xi , i ∈ {2, . . . , d}; 3. multivariate left tail decreasing (MLTD) if (Xi1 , . . . , Xid ) is LTDS for all permutation (i1 , . . . , id ) of (1, . . . , d). Proposition 3.3 Suppose a multivariate copula C(u1 , . . . , ud ) has a lower tail order κL (C), then κL (C) ≥ 1. Moreover, 1. if C is (ultimately) positive lower orthant dependent (PLOD), then κL (C) ≤ d; 2. for any S1 ⊂ S2 ⊆ Id with |S1 | ≥ 2, κL (CS2 )−κL (CS1 ) ≥ 0. In particular, if κL (C) = 1, then for any S ⊂ Id with |S| ≥ 2, κL (CS ) = 1; if C is multivariate left tail decreasing (MLTD), then κL (CS2 ) − κL (CS1 ) ≤ |S2 | − |S1 |. Analogous results hold with κL replaced by κU , and conditions of positive upper orthant dependence and multivariate right tail increasing. 49  ≤  Remark 3.2 The above result says that when some regularity condition holds, marginality will keep the order of tail orders in the sense that margins have smaller tail orders. However, marginality does not inherit the inequality between tail orders of lower and upper tails. For example, take the trivariate Archimedean copula with the ψ function in Example 3.6 in Section 3.3.1 below. Then 3α > 1 + α for 0 < α < 1 so that κL (C) > κU (C) (see Table 3.1). But 2α < 1 + α for 0 < α < 1 so that for the bivariate margins, κL (CI ) < κU (CI ) with |I| = 2. Sometimes partial derivatives and the density have a simpler form than the copula cdf. We hope to know what tail properties will be inherited if we take partial derivatives of the copula. For example, for the lower tail, if C(uw1 , . . . , uwd ) ∼ uκ (u)b(w1 , ..., wd ),  u → 0+ ,  then we want to differentiate both sides of the above with respect to the wj ’s to get: u  ∂C(uw1 , . . . , uwd ) ∂b(w1 , ..., wd ) ∼ uκ (u) , ∂wj ∂wj  u → 0+ ,  and higher order derivatives up to: ud  ∂ d b(w1 , ..., wd ) ∂ d C(uw1 , . . . , uwd ) ∼ uκ (u) , ∂w1 · · · ∂wd ∂w1 · · · ∂wd  u → 0+ .  A sufficient condition is ultimate monotonicity of partial derivatives of the copula (eg: ∂C/∂uj is ultimately monotone in uj at 0+ , and similar conditions are sufficient for higher orders). A proof is similar to that in Theorem 1.7.2 (Monotone density theorem) in Bingham et al. (1987). As an example of using the density to get the tail order, consider a multivariate Gaussian copula with positive definite correlation matrix Σ which satisfies CΦ (u1d ) ∼ uκ (u) = uκ (− log u)ζ , u → 0+ . Then (as can be shown directly with the monotone density theorem), this would be equivalent to cΦ (u1d ) ∼ huκ−d (− log u)ζ , u → 0+ , where h is a constant. Thus, with φd for the multivari-  50  ate Gaussian density, φd Φ−1 (u)1d ; Σ cΦ (u1d ) = lim u→0+ huκ−d (− log u)ζ u→0+ φd (Φ−1 (u))uκ−d (− log u)ζ h φd (z1d ; Σ) = lim d z→−∞ φ (z)[Φ(z)]κ−d [− log(Φ(z))]ζ h φd (z1d ; Σ) = lim κ z→−∞ φ (z)|z|d−κ [− log(φ(z)/|z|)]ζ h  1 = lim  (3.7)  Since the exponent terms dominate the numerator and denominator of (3.7), to cancel the exponent terms, a necessary condition is that κ = 1d Σ−1 1T d , which turns out to be the tail order of the copula CΦ . Also, to cancel the term of |z| in (3.7), we need that d − κ + 2ζ = 0, so ζ = (κ − d)/2.  3.3  Intermediate tail dependence : Archimedean copulas  Archimedean copulas are reflection asymmetric except for the bivariate Frank copula, and have a variety of tail behavior. In this section, we will study the upper/lower tail orders and tail order functions for Archimedean copulas. A new family of one-parameter Archimedean copulas, that interpolates independence and comonotonicity, will be given. This copula possesses intermediate upper and lower tail dependence, and has patterns of tail orders different from existing parametric families. Before getting to the main results, we provide some intuition on conditions on ψ for intermediate upper and lower tail dependence for Cψ . Let G1 , . . . , Gd be univariate cdfs. For η > 0, Gη1 , . . . , Gηd are cdfs, and η is called a resilience parameter. As the parameter η → 0, then random variables with distributions Gη1 , . . . , Gηd tend towards the lower endpoint of support of G1 , . . . , Gd , and as η → ∞, random variables with distributions Gη1 , . . . , Gηd tend towards the upper endpoint of support of G1 , . . . , Gd . There is also a parallel for survival functions and frailty, where the conclusions are reversed when the frailty parameter goes to 0 or ∞. In this way, an Archimedean copula Cψ has a mixture representation with LT ψ. That is, Cψ (u1 , . . . , ud ) =  ∞ 0  the resilience random variable H,  η d j=1 Gj (uj ) dFH (η), where G(u) = exp{−ψ −1 (u)} (0  51  FH is the cdf of ≤ u ≤ 1), and  ψ(s) = ψH (s) =  ∞ −sη dFH (η). 0 e  The mixture means that: there are random  variables X1 , . . . , Xd such that given H = η, they are conditionally independent with respective cdfs Gη1 , . . . , Gηd . If the random variable H has heavy tail at ∞, then there is a “chance” that H = η is large and hence conditionally, X1 , . . . , Xd are all close to their upper endpoints of support (i.e., dependence in the upper tail). Hence conditions on the heaviness of the upper tail of the distribution of H lead to intermediate upper tail dependence. For the opposite tail, if the random variable H has concentration of density near 0, then there is a “chance” that H = η is near zero and hence conditionally, X1 , . . . , Xd are all close to their lower endpoints of support (i.e., dependence in the lower tail). Hence conditions on the density of the lower tail of the distribution of H lead to intermediate lower tail dependence.  3.3.1  Laplace transform and univariate tail heaviness  In this subsection, we relate the asymptotic behavior of a LT to the maximal moment of the positive random variable with the given LT. Definition 3.4 For a positive random variable Y with LT ψ, the maximal nonnegative moment is MY = Mψ = sup{m ≥ 0 : E(Y m ) < ∞}.  (3.8)  MY is 0 if no moments exist and MY is ∞ if all moments exist. A smaller value of MY means that Y has a heavier tail at ∞. The next lemma shows that Mψ is related to the behavior of ψ at 0 when 0 < Mψ < 1. The result for a general non-integer Mψ such that k < Mψ < k + 1 will be derived subsequently. Lemma 3.4 Suppose ψ(s) is the LT of a positive random variable Y , with 0 < MY < 1. If 1 − ψ(s) is regularly varying at 0+ , then 1 − ψ(s) ∈ RVMY (0+ ). Remark 3.3 Even if E(Y ) = ∞, we may also have MY = 1. However, Lemma 3.4 does not hold in general for this case. Remark 3.4 If we write 1 − ψ(s) = sM (s) and (s) → h1 with 0 < h1 < ∞ as s → 0+ , then clearly ψ(s) = 1 − h1 sM + o(sM ), s → 0+ . 52  Proposition 3.5 Suppose ψ(s) is the LT of a positive random variable Y , with k < MY < k + 1 where k ∈ {0} ∪ N+ . If ψ (k) (0) − ψ (k) (s) is regularly varying at 0+ , then ψ (k) (0)−ψ (k) (s) ∈ RVMY −k (0+ ). In particular, if the slowly varying component is (s) and lims→0+ (s) = hk+1 with 0 < hk+1 < ∞, then s → 0+ , ψ(s) = 1 − h1 s + h2 s2 − · · · + (−1)k hk sk + (−1)k+1 hk+1 sMY + o(sMY ), (3.9) where 0 < hi < ∞ for i = 1, . . . , k + 1. The above results can be summarized as follows. If Mψ = ∞, the LT ψ(s) has an infinite Taylor expansion about s = 0. If Mψ is finite and non-integer-valued, then with some regularity conditions, ψ(s) has a Taylor expansion about s = 0 up to order [Mψ ], and the next term after this has order Mψ .  3.3.2  Upper tail  Based on the results in Section 3.3.1, we derive upper tail orders and corresponding tail order functions of multivariate Archimedean copulas; the results extend those of Charpentier and Segers (2009). Proposition 3.6 Let ψ be the LT of a positive random variable and assume that ψ satisfies the condition of Proposition 3.5. Assume that k < Mψ < k + 1 with some k ∈ {1, . . . , d − 1}, then the Archimedean copula Cψ in (2.4) has upper intermediate tail dependence. The corresponding tail order is κU = Mψ . If ψ (i) (0) is finite for all i ∈ Id , then the upper tail order κU = d. If ψ (0) is infinite and 0 < Mψ < 1, then the upper tail order is κU = 1, and particularly for the bivariate case, λU = 2 − 2Mψ . Remark 3.5 If we know the value of a in (3.22) in the proof of Proposition 3.6, then the tail order parameter is lim C ψ (u, u)/(1 − u)Mψ = 2a[−ψ (0)]−1−M (2M − 1)/(1 + M )  u→1−  = a[−ψ (0)]−Mψ (2Mψ − 2)/Mψ .  53  Example 3.6 Consider the LT of Example 4.2 in Joe and Ma (2000) with parameter 0 < α < 1 (see Joe-Ma in Table 1). We refer to this as the normalized integral of the positive stable LT. Note that m = ψ (0) = −1/Γ(1 + α−1 ) is finite, ψ (0) = ∞ and ψ(s) ∼ 1 − s/Γ(1 + α−1 ) as s → 0+ . We can write ψ(s) = 1 + ψ (0)s + o(s), s → 0+ . Let g(s) = ψ (s) − ψ (0) = (1 − exp{−sα })/Γ(1 + α−1 ) ∼ sα /Γ(1 + α−1 ), s → 0+ . Then clearly, g(s) ∈ RVα (0+ ) and can be written as g(s) = sα (s) with (s) → 1/Γ(1 + α−1 ) as s → 0+ . So g(s) = asα + o(sα ), s → 0+ , with a = 1/Γ(1 + α−1 ) > 0. By Proposition 3.6, the copula Cψ has intermediate upper tail dependence when 0 < α < 1. Also, κU = 1 + α and lim  u→0+  C ψ (1 − u, 1 − u) 2[Γ(1 + α−1 )]α (2α − 1) = . u1+α 1+α  It can be shown numerically that the d-variate Archimedean copula with this 1parameter LT family is decreasing in concordance as α increases. As α → 1− , numerically, the limit is close to the independence copula; as α → 0+ , the limit is close to the comonotonic copula. In the next proposition, we state a result for the upper tail order function of Archimedean copulas. Proposition 3.7 Let Cψ be a multivariate Archimedean copula with κU = Mψ being a non-integer in the interval (1, d), and suppose ψ satisfies the condition of Proposition 3.5. With the notation M = Mψ − [Mψ ] and k = [Mψ ], the upper tail order parameter is λU (Cψ ) =  Mh [−ψ  (0)]Mψ  k j=0 (Mψ  − j) ∅=I⊂I  (−1)|I|+k+1 |I|Mψ , d  and the upper tail order function is b∗ (w) =  |I| ∅=I⊂Id (−1)  i∈I  |I| I ∅=I⊂Id (−1)  wi Mψ  Mψ  ,  where h = lims→0+ (s) with |ψ (k) (s) − ψ (k) (0)| = sM (s) as s → 0+ . 54  Remark 3.6 For a d-variate Archimedean copula, the pattern of the upper tail order function also depends on the upper tail order κ. For example, in d = 3, the homogeneous function b∗ is positively proportional to −w1κ −w2κ −w3κ +(w1 +w2 )κ +(w1 +w3 )κ +(w2 +w3 )κ −(w1 +w2 +w3 )κ , 1 < κ < 2; w1κ +w2κ +w3κ −(w1 +w2 )κ −(w1 +w3 )κ −(w2 +w3 )κ +(w1 +w2 +w3 )κ , 2 < κ < 3. The signs of all terms depend on whether 1 < κ < 2 or 2 < κ < 3. The pattern of alternating signs extends to d > 3. This pattern, together with Lemma 3.12, also shows why we don’t have a general form of the tail order function when Mψ is a positive integer. Recently, we have noticed that Larsson and Neˇslehov´a (2011) uses survival d  copulas for a random vector X := (X1 , . . . , Xd ) = RSd as another representation for Archimedean copulas, where R is a radial random variable, the random vector Sd is uniformly distributed on the simplex {(x1 , . . . , xd ) ∈ Rd+ :  d i=1 xi  = 1},  and R and Sd are independent. In parallel to elliptical copulas that are constructed based on random vectors that have a similar scale mixture representation, the authors find that the tail behavior of the radial random variable R will affect the tail dependence patterns of corresponding Archimedean copulas. This representation provides an alternative way to study the dependence in the tails of Archimedean copulas.  3.3.3  Lower tail  For intermediate lower tail dependence of Archimedean copulas, a general result has been obtained in Theorem 3.3 of Charpentier and Segers (2009). We will derive a more concrete and usable result that involves the slowly varying function , and give an interpretation in terms of the (resilience) random variable H which has LT ψ. The condition below on the LT ψ(s) as s → ∞ covers almost all of the LT families in the Appendix of Joe (1997), as well as other LT families that can be  55  obtained by integration or differentiation. Suppose ψ(s) ∼ T (s) = a1 sq exp{−a2 sr } and ψ (s) ∼ T (s),  s → ∞, with a1 > 0, a2 ≥ 0,  (3.10)  where r = 0 implies a2 = 0 and q < 0, and r > 0 implies r ≤ 1 and q can be 0, negative or positive. Note that r > 1 is not possible because of the complete monotonicity property of a LT. The condition can be interpreted as follows. As ψ(s) decreases to 0 more slowly as s → ∞, then the random variable H with LT ψ has a heavier “tail” at 0. Let z = limη→0 fH (η) ∈ [0, ∞], where fH is the density of H and is assumed well-behaved near 0. As z increases, then the “tail” at 0 is heavier. If z = 0, then the tail is lighter as the rate of decrease to 0 is faster. If z = ∞, then the tail is heavier as the rate of increase to ∞ is faster. In terms of the LT and the condition in (3.10), as r increases (with fixed q), the tail of H at 0 gets lighter, and as q increases (with fixed r), the tail of H at 0 gets heavier. The next proposition shows that lower tail dependence behavior is influenced by r. Proposition 3.8 Suppose a LT ψ satisfies the condition in (3.10) with 0 ≤ r ≤ 1. If r = 0, then Cψ has lower tail dependence or lower tail order is 1. If r = 1, then κL (Cψ ) = d. If 0 < r < 1, then Cψ has intermediate lower tail dependence with ζ r 1 < κL (Cψ ) = dr < d, (u) = dq a11−κ a−ζ 2 (− log u) with ζ = (q/r)(1 − d ),  and the tail order function is b(w) =  d dr−1 . i=1 wi  Remark 3.7 Condition (3.10) does not cover all possibilities. It is possible that as s → ∞, ψ(s) goes to 0 slower than anything of form (3.10). Examples are given by LT families LTF and LTG in Joe (1997), leading to Archimedean families such that limu→0+ Cψ (u1d )/u = 1 (for the bivariate case, see families BB2 and BB3 in Joe and Hu (1996), and Joe (1997)). Note that, for LTF, ψ(s) = [1 + δ −1 log(1 + s)]−1/θ with δ > 0 and θ > 0 and as s → ∞, ψ(s) ∼ δ 1/θ (log s)−1/θ ; for LTG, ψ(s) = exp{−[δ −1 log(1 + s)]1/θ } with δ > 0, θ > 1 and as s → ∞, ψ(s) ∼ exp{−δ −1/θ (log s)1/θ }. We refer to Chapters 4 and 5 for relevant research on LTF and LTG. 56  Remark 3.8 Consider the pair (φ, ψ) of LTs where (a) φ (0) is finite and ψ(s) = φ (s)/φ (0) or (b)  ∞ 0 ψ(v)dv  is finite and φ(s) =  ∞ ∞ s ψ(v)dv/ 0 ψ(v)dv.  For  the upper tail, we get Mψ = Mφ − 1 so that LT ψ has a heavier tail and κU (Cψ ) is smaller (stronger intermediate tail dependence) if κU (Cφ ) < d. Proposition 3.8 implies that κL (Cψ ) = κL (Cφ ). But the second level of tail dependence ζ strength comes from the slowly varying function (u) = dq a1−κ a−ζ 1 2 (− log u) .  Since Cψ (u1d ) ∼ uκ (u), u → 0+ , a smaller κ means stronger intermediate lower tail dependence at the first level, and a faster (u) → +∞ or a slower (u) → 0+ means stronger intermediate lower tail dependence at the second level. For the LT tail, a1 sq exp(−a2 sr ), a smaller r means slower decrease to 0 as s → +∞ and the resilience random variable has more probability near 0 and Cψ has more dependence in the lower tail. This can be shown by a smaller tail order dr . A larger q means slower decrease to 0 as s → +∞, which also implies more lower tail dependence. This is seen from a faster increase of (u) → +∞ as u → 0+ when q < 0 and increases, or a slower decrease of (u) → 0+ as u → 0+ when q > 0 and increases. Note that when u is small enough, (− log u)ζ dominates (u).  3.3.4  A new parametric Archimedean copula  By applying the LT of the inverse Gamma distribution, we present a new oneparameter Archimedean copula that exhibits intermediate upper and lower tail dependence, and have essentially a full range of positive dependence from independence to comonotonicity. Example 3.7 (Archimedean Copula based on the Laplace Transform of Inverse Gamma (ACIG)) Let Y = X −1 have the inverse Gamma (IΓ) distribution, where X ∼ Gamma(α, 1) for α > 0. Then it is straightforward to derive that MY = α. The LT of the inverse Gamma distribution: ψ(s; α) =  √ 2 α/2 s Kα (2 s), Γ(α)  s ≥ 0, α > 0,  (3.11)  where Kα is the modified Bessel function of the second kind. (Please see Section 3.6 for the derivation of (3.11).) It can be shown numerically that the d-variate Archimedean copula with this 1-parameter LT family is decreasing in concordance 57  as α increases, with limits of the independence copula as α → ∞ and the comonotonic copula as α → 0. Proposition 3.9 Let Cψ be an Archimedean copula constructed by (3.11). If α ∈ (0, +∞) is not an integer, then the upper tail order is max{1, min{α, d}}. The √ lower tail order is d. Remark 3.9 For the bivariate case, κU = max{1, min{α, 2}} and κL =  √  2.  Hence there is reflection asymmetry with skewness to the upper tail for 0 < α < √ √ 2 and skewness to the lower tail for α > 2 . Contour plots of this new Archimedean copula are given in Figure 3.1. Figure 3.1: Contour plots: ACIG copula + standard normal margins α=4  α=2  0.06  0.02  2  0.02  2  0.06  0.1  0.12  1  1  1  0.06 0.1  0.  0  0  0.1  2  −1  0.1  −1  0.08  0.04  −1  0  1  2  −2  −1  0  1  2  2  2  2  4  0  4  0.  0. 1  2  0.1  −1  0 −1  0.16  0.16  0.1  2  0.06  −2  −2  −2  8 0.0  0.06  1  0.02  −2  −1  2  0.04  1  1  1 0  0.12  0  1  0.1  2 0.1  0.02  −1  0  α = 0.5  0.08  0.14  −1  0.08  0.06  −2  −1  0.04  0.04 0.1  −2  α = 0.8  α=1  2  0.04  −2  −2  0.04  −2  −2  0.1  0.08  0. 1  −1  0.08  0.16 0.14  0  14  16  2  0.02  α = 1.414214  0  1  2  0.02  −2  −1  0  1  2  To conclude this subsection, we list the tail orders for some Archimedean copulas that interpolate independence and comonotonicity in Table 3.1. A variety of tail behavior obtains from known parametric Archimedean families and the new Archimedean family. Note that the bivariate Frank copula is reflection symmetric. But for d-dimensional Frank copula with d ≥ 3, it can be shown numerically 58  that C ψ ( 12 1d ) > Cψ ( 12 1d ) for parameters θ > 0, although the lower tail order and upper tail order are the same. Some of the results in this table can be found in Charpentier and Segers (2009) and Heffernan (2000). For all of the examples in Table 3.1, the upper and lower tail orders decrease or remain constant as the dependence parameter(s) leads to increased dependence/concordance. Table 3.1: Tail order of some Archimedean copulas that interpolate independence and comonotonicity Copula/LT family κL −1 −θ −s Frank; log-series LT −θ log[1 − (1 − e )e ] (θ > 0) d 1 MTCJ; gamma LT (1 + s)−1/θ (θ > 0) 1 −s 1/θ Joe; Sibuya LT 1 − (1 − e ) (θ > 1) d Gumbel; positive stable LT exp{−s1/θ } (θ > 1) d1/θ Joe-Hu; BB1 extension; (1 + s1/δ )−1/θ (θ > 0, δ > 1) 1 Joe-Hu; BB7 extension; 1 − [1 − (1 + s)−1/δ ]1/θ (θ > 1, δ > 0) 1 Crowder; BB9 extension; exp{−(αθ + s)1/θ + α} (θ > 1) d1/θ ∞ −v α −1 dv/Γ(1 + α ) (0 < α < 1) Joe-Ma; s e dα √ ACIG; LT of inverse gamma 2Γ−1 (α)sα/2 Kα (2 s ) (α > 0) d1/2  κU d d 1 1 1 1 d 1+α (d ∧ α) ∨ 1  1. Mardia-Takahasi-Cook-Johnson, (see Cook and Johnson, 1981).  3.4  Intermediate tail dependence: Mixture of max-id copulas  As an extension of Archimedean copulas, we study in this section the tail orders for copulas that are constructed with mixtures of max-id copulas. Some results studied in Joe (1997) are extended to intermediate tail dependence. Let F be a d-variate cdf. If F t is also a cdf function for all t > 0, then F is max-id (Joe and Hu, 1996). The class of copulas based on mixture of max-id distributions has led to interesting classes of bivariate two-parameter copula families with both upper and lower tail dependence (e.g., labeled as BB1, BB4, BB7 in Joe (1997)). As well, other forms of intermediate tail dependence behavior are possible. These types of copulas will give us more flexibility in choices of bivariate linking copulas in vines (Aas et al., 2009; Joe et al., 2010). 59  Here we generalize Theorems 4.13 and 4.16 in Joe (1997) to multivariate versions and intermediate tail dependence. In the earlier research on copulas, the analyses determined when tail dependence (tail order κ = 1) can occur for different copula families; in that setting, the tail order occurred within the sufficient condition in Theorem 4.16 of Joe (1997). Let K be a multivariate max-id copula and ψ be a LT of a positive random variable, and consider the copulas that are of the following form C(u1 , . . . , ud ) = ψ − log K e−ψ  −1 (u ) 1  , . . . , e−ψ  −1 (u ) d  (3.12)  Proposition 3.10 Suppose that a copula C be constructed by (3.12). 1. If ψ satisfies the condition of Proposition 3.5 with some k ∈ {1, . . . , d − 1} and κU (KI ) > 1 for any marginal copula KI , then C has upper intermediate tail dependence and κU (C) = κU (Cψ ). 2. If 1 − ψ(s) ∈ RVβ (0+ ), κU (K) = 1 with marginal copula KI (u1|I| ) ∼ u I (u), u → 0+ such that limu→0+ |J|−1 h J ∅=J⊂I (−1)  I (u)  = hI ∈ (0, 1], 0 < h∗I =  |I|−1 (h∗ )β ∅=I⊂Id (−1) I |I|−1 (h∗ )β . (−1) ∅=I⊂Id I  ≤ 1 and 0 <  κU (C) = 1 with λU (C) =  ≤ 1, then  Proposition 3.11 Suppose that a copula C be constructed by (3.12) with 1 ≤ α = κL (K) ≤ d. If −ψ(s)/ψ (s) ∈ RVβ with 0 < β ≤ 1, and 1 < α1−β < d, then the copula C has lower intermediate tail dependence κL (C) = α1−β , and κL (C) = ξ(α, β) · κL (Cψ ) with ξ(α, β) = (α/d)1−β ∈ (0, 1]. Also, κL (K) = 1 implies that κL (C) = 1. Remark 3.10 Note that κL (C) is less than or equal to both κL (K) and κL (Cψ ). K can be the independence copula or have intermediate lower tail dependence. The lower tail order of the copula C is increasing in κL (K). One consequence of Propositions 3.10 and 3.11 is that if κU (K) = d and κL (K) = d then κU (C) = κU (Cψ ) and κL (C) = κL (Cψ ). Hence if K is chosen as the parametric Frank copula family with parameter θ ≥ 0, then C(u1 , ..., ud ; θ) as given in (3.12) will be increasing in concordance as θ increases. The parameter θ affects dependence only, while the LT ψ controls the upper and lower tail orders. 60  When we take K as the independence copula or the Frank copula with positive dependence, and the LT has tail of the form ψ(s) ∼ a1 sq exp{−a2 sr }, s → ∞, 0 ≤ r < 1, where a1 , a2 are some positive constants, then we can construct a new family of Archimedean copulas that satisfies the condition of Proposition 3.11. In dimensions d ≥ 3, Archimedean and mixture of max-id copula families cannot achieve the range of dependence available from vine copulas (Bedford and Cooke, 2001; Aas et al., 2009; Joe et al., 2010). But for d = 2, the mixture of maxid approach can lead to more candidates, with a variety of upper and lower tail behavior, to be used as bivariate linking copulas in vines. For instance, from Table 3.1, the preceding subsections and propositions, the Joe-Ma ψ function, which is the normalized integral of the positive stable LT, combined with the bivariate Gaussian copula with  ≥ 0 can lead to a two-parameter family with more flexible  upper and lower tail orders. Note that, the bivariate Gaussian density is Totally Positive of order 2 if  3.5  ≥ 0, and hence max-id (Joe, 1997, Theorem 2.6).  Discussion  We have shown how the concept of tail order is useful to quantify the strength of dependence in the upper and lower tails, as well as the direction of reflection asymmetry. One- and two-parameter families that are Archimedean copulas and mixture of max-id copulas together can cover a wide range of tail orders. The interpretation through the latent resilience variable shows why Archimedean copulas can obtain a full range of tail orders by varying the density of the resilience at 0 and ∞. In order to get our results for Archimedean copulas, we needed Proposition 3.5 which, on its own, contributes knowledge about LTs. Archimedean copulas only have exchangeable dependence but their bivariate versions can be used within vines. Vine copulas (Bedford and Cooke, 2001; Aas et al., 2009; Joe et al., 2010) in dimension d, which include multivariate Gaussian and t copulas as special cases, are built from d(d − 1)/2 bivariate linking copulas, of which d − 1 are bivariate marginal copulas and the remainder are conditional bivariate copulas with the number of conditioning variables between 1 to d − 2. By choosing bivariate linking copulas with flexible tail orders and reflecting symmetry/asymmetry, we can get vine copulas to cover a wide range of tail behavior,  61  as well as dependence structures. The study of tail orders of vine copulas in terms of the tail orders of the bivariate linking copulas will be studied in future research. For vine copulas, we are also interested in conditions that retain consistent relation of upper and lower tail orders for all margins.  3.6  Proofs  Derivation of LT of the inverse Gamma distribution: With Y = X −1 and X ∼ Gamma(α, 1), the LT is derived as ∞  ψ(s) = ψ(s; α) = E(e−sY ) = E(e−s/X ) = [Γ(α)]−1  e−s/x xα−1 e−x dx.  0  From the GIG(ν, χ, ϕ) density (McNeil et al., 2005), ∞ 0  √ wν−1 exp{− 21 (χw−1 + ϕw)} dw = 2(χ/ϕ)ν/2 Kν ( χϕ ).  Note that Kν = K−ν . Hence with χ = 2s, ϕ = 2, ν = α √ √ ψ(s; α) = 2Γ−1 (α)(2s/2)α/2 Kα ( 2s · 2 ) = 2Γ−1 (α)sα/2 Kα (2 s ).  Proof of Proposition 3.1: Letting r := F −1 (1 − u) and b = Example 6.2 (i) of Hashorva (2007), as u →  0+  2/(1 + ), then by  and thus r → ∞,  C(1 − u, 1 − u) = P X1 > F −1 (1 − u), X2 > F −1 (1 − u) = P[X1 > r, X2 > r] = (1 + o(1))  (1 − 2 )3/2 [a(b r)/r][1 − FR (b r)], 2π(1 − )2  (3.13)  where FR is the cdf of R and a(·) is an auxiliary function of R with respect to the Gumbel Maximum Domain of Attraction in the sense of (2.10). As u → 0+ , i.e., r → ∞, both a(b r)/r → 0 (see Theorems 3.3.26 and 3.3.27 of Embrechts et al. (1997)) and 1 − FR (b r) → 0. Let G(x) := 1/[1 − FR (x)], then G : R → R+ is increasing and the condition of (2.10) is equivalent to that G ∈ Γ-varying with 62  auxiliary function a(·) (de Haan, 1970, Definition 1.5.1). The inverse function of a Γ-varying function is a Π-varying function (de Haan, 1974, Corollary 1.10). Therefore, G−1 ∈ Π-varying. Assuming that an auxiliary function of G−1 is a0 (·), by Lemma 1.2.9 of de Haan and Ferreira (2006), the auxiliary function a0 (·) of the Π-varying function G−1 is slowly varying at ∞. Moreover, a0 (t) = a(G−1 (t)) (de Haan, 1974, Corollary 1.10). So, a(x) = a0 (G(x)). Then in (3.13), a(b r)/r = a0 (G(b r))/r = a0 (1/[1 − FR (b r)])/r, while 1 − FR (b r) is rapidly varying in r at ∞ due to the fact that G is Γ-varying and any Γ-varying function is rapidly varying (de Haan, 1970, Theorem 1.5.1). Therefore, 2/(1 + )F −1 (1 − u)  1 − FR (b r) = 1 − FR  dominates the tail behavior of (3.13) as u → 0, and thus determines the corresponding tail order of the elliptical copula. By the definition of tail order in Definition 3.1, we may also obtain the upper tail order by the following κ = lim  u→0+  log (1 − FR (b r)) log C(1 − u, 1 − u) = lim . r→∞ log (1 − F (r)) log u  By Example 6.2 (iii) of Hashorva (2007), as r → ∞, P[X1 > r] = (1 + o(1))(2π)−1/2 [a(r)/r]1/2 [1 − FR (r)]. Due to the similar argument as before, 1 − FR (r) dominates the tail behavior of P[X1 > r], as r → ∞. Therefore, we may write κ = lim  r→∞  log (1 − FR (b r)) , log (1 − FR (r))  which completes the proof. Proof of Proposition 3.3: Assuming C(u1d ) ∼ uκL (C) (u), u → 0+ , with (u) ∈ RV0 (0+ ), for any copula C and 0 ≤ u ≤ 1, C(u1d ) ≤ u. Therefore, κL (C) ≥ 1.  63  To prove (1), by the condition of PLOD, we have C(u1d ) ≥ ud for any 0 ≤ u ≤ 1 and thus, κL (C) ≤ d. To prove (2), choosing S1 ⊂ S2 with |S2 | − |S1 | = j ∈ N+ . Let us consider the case where j = 1 first, for some l ∈ {1, . . . , |S2 |} and any 0 ≤ u ≤ 1, CS2 (u1|S2 | ) = P[Ul ≤ u|U1 ≤ u, . . . , Ul−1 ≤ u, Ul+1 ≤ u, . . . , U|S2 | ≤ u] × P[U1 ≤ u, . . . , Ul−1 ≤ u, Ul+1 ≤ u, . . . , U|S2 | ≤ u] ≥ P[Ul ≤ u] × P[U1 ≤ u, . . . , Ul−1 ≤ u, Ul+1 ≤ u, . . . , U|S2 | ≤ u] = uCS1 (u1|S1 | ). The inequality is due to the MLTD of C. Clearly, κL (CS2 ) − κL (CS1 ) ≤ 1. Since P[Ul ≤ u|U1 ≤ u, . . . , Ul−1 ≤ u, Ul+1 ≤ u, . . . , U|S2 | ≤ u] ≤ 1, CS2 (u1|S2 | ) ≤ CS1 (u1|S1 | ) and thus, κL (CS2 ) − κL (CS1 ) ≥ 0. An iterated argument will prove the case for a general j: 0 ≤ κL (CS2 ) − κL (CS1 ) ≤ |S2 | − |S1 |. If κL (C) = 1, then for any S ⊂ Id with |S| ≥ 2, we have 1 ≤ κL (CS ) ≤ κL (C) = 1, which completes the proof. For κL (C) = 1, note that the MLTD condition is not needed.  Proof of Lemma 3.4: Let Z be an exponential random variable, independent of Y , with mean 1. Choose any fixed m with 0 < m < 1. Then E(Z −m ) = Γ(1 − m), and if we define Wm = (Y /Z)m , then for any w > 0, P [Wm ≥ w] = P[Z ≤ Y w−1/m ] ∞  =  1 − exp{−yw−1/m } FY (dy) = 1 − ψ(w−1/m ),  0  where FY is the cdf of Y . Therefore, E(Y m ) < ∞ implies E(Wm ) < ∞ and limw→∞ w[1 − ψ(w−1/m )] = 0, i.e., lim [1 − ψ(s)]/sm = 0.  s→0+  (3.14)  If 1 − ψ(s) is regularly varying at 0+ , then we can write 1 − ψ(s) = sα (s) with α = 0, where (s) ∈ RV0 (0+ ). Then, (3.14) implies that lims→0+ sα−m (s) = 0. Let > 0 be arbitrarily small. If m = MY − , then we have E(Y MY − ) < ∞ and 64  thus lims→0+ sα−MY + (s) = 0. Therefore, α ≥ MY − . Also by a result on page 49 of Chung (1974), (3.14) implies that for any 0 < δ < 1, E Y m(1−δ) < ∞. If we assume that there exists an m = MY + such that, lims→0+ E Y (MY + E Y  )(1−δ)  sα−MY −  > 0 with  (s) = 0, then for any small δ > 0,  < ∞. Then we may choose some δ < /( + MY ), and get  (MY + )(1−δ )  < ∞ with (MY + )(1 − δ ) > MY , which gives rise to a  contradiction. Thus, for any > 0, we must have lims→0+ sα−MY − (s) = 0, and hence, α − MY − ≤ 0. So, MY − ≤ α ≤ M Y + , which completes the proof. Proof of Proposition 3.5: This proof extends that in Lemma 3.4, which corresponds to the case where k = 0. For a positive integer j, let Zj ∼ Gamma(j + 1, 1) so that E(Zj−m ) = Γ(j + 1 − m)/Γ(j + 1) if 0 < m < j + 1. Let Wm,j = (Y /Zj )m where Y is independent of Zj . Then for 0 < m < j + 1, E(Wm,j ) < ∞ if and only if E(Y m ) < ∞. Next, similar to the proof of Lemma 3.4, if Y has LT ψ and moments up to order k, for j ∈ {0, 1, . . . , k} and 0 < m < j + 1, ∞  Pr Wm,j = (Y /Zj )m ≥ w = Pr(Zj ≤ Y w−1/m ) =  FZj (yw−1/m ) dFY (y)  0 j  ∞  1−  = 0  y i w−i/m  i=0 j  =1− i=0  i!  exp{−yw−1/m } dFY (y)  w−i/m (−1)i ψ (i) (w−1/m ). i!  Suppose 0 < m < min{j + 1, MY }. Then E(Y m ) < ∞ implies that j  w 1− i=0  w−i/m (−1)i ψ (i) (w−1/m ) → 0, i!  65  w → ∞,  i.e., j  s−m 1 − i=0  si (−1)i ψ (i) (s) → 0, i!  s → 0+ .  (3.15)  Assuming ψ has derivatives at zero up to kth order, then for positive integer j ≤ k, the main term in (3.15) is j  1− i=0  si (−1)i ψ (i) (s) i! j−1  i=0 j−1  si  =1−  i!  i=0 j  j−i  si (−1)i i!  =1−  sl ψ (i+l) (0)/l! + o(sj−i ) − l=0 j  sl−i ψ (l) (0)/(l − i)! −  (−1)i l=i  sl ψ (0) l!  l∧(j−1)  (l)  =1− l=0  = 1 − ψ(0) − ψ (j) (0) = (−1)j−1  sj j!  (−1)i i=0 j s  j!  sj (−1)j ψ (j) (s) j!  sj (−1)j ψ (j) (s) + o(sj ) j!  l! sj − (−1)j ψ (j) (s) + o(sj ) i!(l − i)! j!  −(−1)j −  sj (−1)j ψ (j) (s) + o(sj ) j!  ψ (j) (s) − ψ (j) (0) + o(sj ).  (3.16)  Hence (3.15) implies that sj−m ψ (j) (s) − ψ (j) (0) → 0,  s → 0+ ,  if j is a non-negative integer less than MY and m < MY . In particular, if k is a non-negative integer such that k < m < MY < k + 1, then sk−m ψ (k) (s) − ψ (k) (0) → 0,  s → 0+ .  If ψ (k) (0) − ψ (k) (s) is regularly varying at 0+ , we write ψ (k) (0) − ψ (k) (s) = sα (s). For any  > 0, a similar argument in the proof of Lemma 3.4 will prove  that α ≥ MY − k − . Now we prove the other direction. We assume that there  66  exists an > 0 with m = MY + such that, lim sα+k−MY − (s) = 0,  (3.17)  s→0+  that is, sk−MY − ψ (k) (s) − ψ (k) (0) → 0 as s → 0. Since ψ is completely monotonic, ψ (k) (0) − ψ (k) (s) is either negative or positive as s → 0+ . That is, (−1)k [ψ (k) (0) − ψ (k) (s)] > 0. The following argument is for an even k, and similar when k is odd. Then by the Karamata’s theorem (Resnick, 2007), regular variation of ψ (k) (0) − ψ (k) (s) implies that − ψ (k−1) (x) + ψ (k−1) (0) + xψ (k) (0) x  =  ψ (k) (0) − ψ (k) (s) ds ∼ (α + 1)−1 xα+1 (x),  x → 0+ .  (3.18)  0  Since  x (k) (0) − ψ (k) (s)]ds 0 [ψ  is again regularly varying, we can take the integra-  tion on both sides repeatedly and obtain for j = 0, 1, . . . , k, j  −ψ  (k−j)  xj−i (k−i) ψ (0) ∼ (j − i)!  (x) + i=0  j  1 α+i  α i=0  xα+j (x),  x → 0+ . (3.19)  Multiplying both sides of (3.19) by  LHSj =: −  ∼  xk−j k−j (k−j)! (−1)  xk−j (−1)k−j ψ (k−j) (x) + (k − j)!  (−1)k−j (k − j)!  j  α i=0  1 α+i  j  i=0  leads to  (−1)k−j xk−i ψ (k−i) (0) (k − j)!(j − i)!  xk+α (x),  x → 0+ .  (3.20)  Then we add the left-hand side of (3.20) for j = 0, . . . , k, and after rearranging the summand, we have k  k  LHSj = − j=0  i=0  xi (−1)i ψ (i) (x) + 1 + i!  k−1 k i=0 j=i  (−1)k−j xk−i ψ (k−i) (0). (k − j)!(j − i)! (3.21)  67  By the binomial theorem, for each given i ∈ (0, . . . , k − 1),  (−1)k−j k j=i (k−j)!(j−i)!  ≡ 0.  Then, from (3.20) and (3.21) we can conclude that k  k  LHSj = 1 − j=0  i=0  xi (−1)i ψ (i) (x) = O xk+α (x) . i!  Therefore, multiplying both sides of the above by s−MY − and using (3.17), k  s  −MY −  1− i=0  si (−1)i ψ (i) (s) → 0, i!  Then for any small δ > 0, E Y (MY + δ < /( + MY ), and get E Y (MY +  )(1−δ)  )(1−δ )  s → 0+ .  < ∞. Then we may choose some < ∞ with (MY + )(1 − δ ) > MY ,  which gives rise to a contradiction to the fact that MY is the maximal moment. Thus, for any  > 0, we must have lims→0+ sα+k−MY − (s) = 0, and hence,  α ≤ MY − k + . Thus, α = MY − k. To prove the last statement of the proposition, since 1 − φ(s) = 1 − ψ (k) (s)/ψ (k) (0) = ψ (k) (0) − ψ (k) (s) /ψ (k) (0) ∈ RVMY −k (0+ ), by Remark 3.4, we have φ(s) = ψ (k) (s)/ψ (k) (0) = 1 − hk+1 sMY −k + o(sMY −k ). Then, by integration, we will have ψ(s) = 1 + ψ (1) (0)s + 12 ψ (2) (0)s2 + · · · + (−1)k+1 hk+1 sk+MY −k + o(sk+MY −k ) = 1 − h1 s + h2 s2 − · · · + (−1)k+1 hk+1 sMY + o(sMY )  s → 0+ ,  where 0 < hi < ∞. The integration is due to Lemma 31 of Breitung (1994). Proof of Proposition 3.6: We provide the proof only for the bivariate case. For d ≥ 3, the intermediate upper tail dependence can be studied analogously, and the (omitted) proof is similar but with more complicated notation. Let ψ (0) = m with −∞ < m < 0, then by Proposition 3.5, as s → 0+ , 68  letting M = Mψ − 1, g(s) = ψ (s) − m = asM + o(sM ),  (0 < a < ∞; 0 < M < 1).  Since g (s) = ψ (s) is increasing as s → 0+ , if we write g(s) ∼ asM (s), s → 0+ , where (s) ∈ RV0 (0+ ) and (s) → 1 as s → 0+ . Note that s  s  g(s) = ψ (s) − m =  g (x)dx.  ψ (x)dx = 0  0  By the Monotone Density Theorem (Bingham et al., 1987, Theorem 1.7.2), ψ (s) = g (s) ∼ aM sM −1 (s),  s → 0+ .  (3.22)  Thus, lims→0+ ψ (2s)/ψ (s) = 2M −1 . Observe that for 0 < ζ < 1, C ψ (u, u) (1 − u)1+ζ 1 − 2u + ψ(2ψ −1 (u)) −2 + 2ψ (2ψ −1 (u))/ψ (ψ −1 (u)) = lim = lim (1 − u)1+ζ −(1 + ζ)(1 − u)ζ u→1− u→1− 4ψ (2ψ −1 (u))/[ψ (ψ −1 (u))]2 − 2ψ (ψ −1 (u))ψ (2ψ −1 (u))/[ψ (ψ −1 (u))]3 = lim ζ(1 + ζ)(1 − u)ζ−1 u→1− 4ψ (2s)/[ψ (s)]2 − 2ψ (s)ψ (2s)/[ψ (s)]3 = lim (letting s = ψ −1 (u)) ζ(1 + ζ)(1 − ψ(s))ζ−1 s→0+ lim  u→1−  = lim  s→0+  = lim  s→0+  −2 4m−2 ψψ (2s) 4m−2 ψ (2s) − 2m−2 ψ (s) (s) − 2m = lim ζ−1 ζ(1 + ζ)(1 − ψ(s))ζ−1 s→0+ ζ(1 + ζ) (1−ψ(s)) ψ (s)  2m−2 (2M − 1) ζ−1  ζ(1 + ζ) (1−ψ(s)) ψ (s)  .  By Proposition 3.5, there is a constant h > 0 such that 1 − ψ(s) = −ms − hsM +1 + o(sM +1 ),  69  s → 0+ .  Then, as s → 0+ , [1 − ψ(s)]ζ−1 ∼ (−m)ζ−1 sζ−1 . In addition, it has been shown that ψ (s) ∼ aM sM −1 (s) as s → 0+ and 2m−2 (2M − 1) is finite. Hence, the intermediate tail dependence exists if and only if ζ = M and κU = 1 + M = Mψ . The proof for the case of k = 0 is similar, by applying Proposition 3.5. Lemma 3.12 Let d ≥ 2 be a positive integer and let j be a positive integer that is less than d. Let j  (−1)|I|−1  Sdj (w1 , . . . , wd ) =  wi  .  (3.23)  i∈I  ∅=I⊂Id  Then Sdj ≡ 0. Proof of Lemma 3.12: When j = 1, by the binomial theorem, for any n ∈ N+ , n i n i=0 (−1) i  = 0, so that Sd1 ≡ 0 for d ≥ 2.  For 1 < j < d, Sdj is a symmetric homogeneous function of order j, and its first order partial derivatives are homogeneous of order j − 1. By recursion with Euler’s formula for homogeneous functions to the jth order partial derivatives Sdj (w) =  =  1 j 1 j!  d i=1  ∂Sdj (w) 1 wi = ∂wi j(j − 1)  d  d  ··· i1 =1  ij =1  ∂j S  dj (w)  ∂wi1 · · · ∂wij  d  d  i1 =1 i2 =1  ∂ 2 Sdj (w) wi wi ∂wi1 ∂wi2 1 2  wi1 · · · wij .  (3.24)  We will show that all the jth order partial derivatives are 0. Because of symmetry, n  we consider only terms for which wi1 . . . wij = w1n1 · · · wp p where 1 ≤ p ≤ j < d,  70  n1 > 0, . . . , np > 0 and n1 + · · · + np = j. Then ∂ j Sdj (w) = j!(−1)p−1 + ∂ n1 w1 · · · ∂ np wp  (−1)p+|J|−1 j! ∅=J⊂{p+1,...,d}  (−1)|J|  = j!(−1)p−1 1 +  ∅=J⊂{p+1,...,d} d−p  = j!(−1)p−1  (−1)i i=0  d−p i  = 0,  which completes the proof. Note that (3.23) is not zero for j = d because (3.24) would include a non-zero term such as ∂ d Sdj /∂w1 · · · ∂wd = (−1)d−1 d!. In fact, there are d! non-zero terms in (3.24) when (i1 , . . . , id ) is a permutation of (1, . . . , d), and Sdd (w1 , . . . , wd ) = (−1)d−1 d!  d i=1 wi .  Proof of Proposition 3.7: Consider  lim  P  u→0+  i∈Id {Ui  ≥ 1 − uwi }  uk+M (−1)|I|−1 lim  = ∅=I⊂Id  P  u→0+  i∈I {Ui ≥ 1 uk+M  − uwi }  .  By Proposition 3.5, since the function w → 1 − ψ(w) ∈ RV1 (0+ ), then we have  71  w → ψ −1 (1 − w) ∈ RV1 (0+ ). Thus, lim  i∈I {Ui ≥ 1 uk+M  P  u→0+  − uwi }  1 − ψ[ψ −1 (1 − uw1 ) + · · · + ψ −1 (1 − uwd )] uk+M u→0+ 1 − ψ[ψ −1 (1 − uw1 ) + · · · + ψ −1 (1 − uwd )] = lim {1 − ψ[ψ −1 (1 − u)]}k+M u→0+ = lim  1 − ψ ψ −1 (1 − u) = lim  −1 ψ −1 (1−uw1 ) d) + · · · + ψψ−1(1−uw ψ −1 (1−u) (1−u) ψ[ψ −1 (1 − u)]}k+M  {1 − 1−ψ − u) i∈I wi = lim −1 + {1 − ψ[ψ (1 − u)]}k+M u→0 u→0+  ψ −1 (1  .  Let s = ψ −1 (1 − u) and (−1)|I|−1 lim  Q(w) =  s→0+  ∅=I⊂Id  1−ψ s i∈I wi {1 − ψ(s)}k+M  .  (3.25)  To obtain the limit in (3.25), we may use the l’Hˆopital’s rule. For the first k derivatives of the numerator, for fixed w and ψ (j) (0) finite for j = 1, . . . , k,  s→0  j  (−1)|I|−1 ψ (j) s  lim  wi  wi  i∈I  ∅=I⊂Id  = 0,  j ∈ {1, . . . , k},  i∈I  because by Lemma 3.12, j  (−1)|I|−1  wi  j ∈ {1, . . . , k}, 1 ≤ k < d.  = 0,  i∈I  ∅=I⊂Id  Then by the l’Hˆopital’s rule (k + 1 applications) (−1)|I|−1 ×  Q(w) = ∅=I⊂Id  × lim  s→0+  −ψ (k+1) [s [−ψ (1) (s)]k+1 {  i∈I k j=0 (k  wi ]  i∈I  wi  k+1  /ψ (k+1) (s)  + M − j)} (1 − ψ(s))M −1 /ψ (k+1) (s)  72  .  Since g(s) = |ψ (k) (s) − ψ (k) (0)| = hsM + o(sM ) with h > 0, we can write g(s) ∼ sM (s), s → 0+ with a slowly varying function (s) → h as s → 0+ . Note that, g(s) =  s (k+1) (x)|dx 0 |ψ  =  s 0 g  (x)dx, and g (s) is monotonic as s → 0+ ,  then by the Monotone density theorem, |ψ (k+1) (s)| = g (s) ∼ M sM −1 (s) and ψ (k+1) (s) ∼ (−1)k+1 M sM −1 (s). Therefore, Q(w) =  Mh [−ψ (1) (0)]k+M  k j=0 (k  + M − j) ∅=I⊂I  k+M  (−1)|I|+k+1  wi  .  i∈I  d  Then, the upper tail order function is b∗ (w) =  Q(w) = Q(1d )  |I| ∅=I⊂Id (−1)  i∈I  |I| I ∅=I⊂Id (−1)  wi  k+M  k+M  .  Note that this is a homogeneous function in w of order κU = Mψ = k + M . This completes the proof . Proof of Proposition 3.8: If r = 0, then ψ −1 (t) ∼ (t/a1 )1/q as t → 0+ (where q < 0). If r > 0, then for large s and small t, in log ψ(s) = log t ∼ log a1 + q log s − a2 sr ,  s → ∞,  the third term dominates, so that ψ −1 (t) ∼ [(− log t)/a2 ]1/r ,  t → 0+ .  Next, consider Cψ (u1d ) = ψ(dψ −1 (u)). −1/q 1/q u )  For r = 0, one gets ψ(dψ −1 (u)) ∼ ψ(da1  ∼ dq u, as u → 0+ , with  dq ∈ (0, 1), so that κL = 1. For 0 < r < 1, suppose ∆L,κ = lim  u→0+  ψ dψ −1 (u) > 0. uκ (− log u)ζ  73  Then by l’Hˆopital’s rule, ∆L,κ = lim  u→0+  dψ dψ −1 (u) /ψ ψ −1 (u) dψ (ds)/ψ (s) = lim . κ−1 ζ s→∞ κu (− log u) κ[ψ(s)]κ−1 [− log ψ(s)]ζ (3.26)  By condition (3.10), the dominating term of ψ (s) or T (s) is ψ (s) ∼ −a1 a2 rsq+r−1 exp{−a2 sr },  s → ∞.  Consider the limit of the right-hand side of (3.26) without the factor d/κ: ψ (ds)/ψ (s) ∼ dq+r−1 exp{−a2 (dr − 1)sr },  s → ∞;  [ψ(s)]κ−1 [− log ψ(s)]ζ ∼ [a1 sq ]κ−1 exp{−a2 (κ − 1)sr } · [a2 sr − q log s − log a1 ]ζ aζ2 sq(κ−1)+rζ exp{−a2 (κ − 1)sr }, ∼ aκ−1 1  s → ∞.  Hence κ = dr , q(κ − 1) + rζ = 0 or ζ = (q/r)(1 − κ) = (q/r)(1 − dr ) and ∆L,κ =  ddq+r−1 aζ2 κaκ−1 1  =  dq aζ2 aκ−1 1  .  ζ a−ζ So (u) = ∆L,κ · (− log u)ζ = dq a1−κ 2 (− log u) . 1  Under the condition in (3.10), it can be verified that −ψ(s)/ψ (s) ∈ RV1−r , which satisfies the condition in Theorem 3.3 of Charpentier and Segers (2009). So the tail order function is obtained. Proof of Proposition 3.9: When 0 < ν < 1, Kν (s) ∼  1 2  Γ(ν) s/2  −ν  + Γ(−ν) s/2  ν  .  (3.27)  We refer to the website of Wolfram Research Bes for asymptotic behavior of modified Bessel function of the second kind. For 0 < α < 1, ψ(s; α) ∼ 1 +  Γ(−α) α s , Γ(α)  74  s → 0+ ,  and 1 − ψ(s; α) ∼ −sα Γ(−α)/Γ(α) ∈ RVα (0+ ). This is consistent with Lemma 3.4. Now, let us consider the case where α is non-integer with α > 1. For an integer j with 0 < j < α and X ∼ Gamma(α − j, 1), ψ (j) (s; α) = [Γ(α)]−1 (−1)j j  = (−1) 2Γ  −1  (α)s  ∞  e−s/x xα−j−1 e−x dx = (−1)j  0 (α−j)/2  √ Kα−j (2 s ).  Γ(α − j) E(e−s/X ) Γ(α)  So, φ(s; α, j) = ψ (j) (s; α)/ψ (j) (0; α) is the LT of Z = 1/X ∼ IΓ(α − j, 1) with MZ = α − j. When ν is non-integer with |ν| > 1, the behavior near 0 of Kν is Kν (x) ∼ x−|ν| 2|ν|−1 Γ(|ν|) 1 +  x2 . 4(1 − |ν|)  So, for integer j < α − 1 (−1)j ψ (j) (s; α) √ s Γ(α − j) = 2Γ−1 (α)s(α−j)/2 Kα−j (2 s ) ∼ 1+ , Γ(α) 1−α+j  s → 0;  for k < α < k + 1, where k ∈ N+ , then by (3.27) (−1)k ψ (k) (s; α) √ Γ(α − k) Γ(−α + k) α−k = 2Γ−1 (α)s(α−k)/2 Kα−k (2 s ) ∼ + s , Γ(α) Γ(α)  s → 0.  Therefore, |ψ (k) (s) − ψ (k) (0)| ∈ RVα−k (0+ ), which is consistent with Proposition 3.5. Then, by Proposition 3.5, there is a positive constant hk+1 such that ψ(s) =1 + ψ (1) (0)s + 12 ψ (2) (0)s2 + · · · + (−1)k ψ (k) (0)sk /k!+ + (−1)k+1 hk+1 sα + o(sα ). The upper tail order of the d-variate Archimedean copula Cψ follows from Propositions 3.6 and 3.7. Therefore, if α ∈ (0, +∞) is not an integer, the upper tail order is max{1, min{α, d}}. 75  Next we investigate the lower tail. From Abramowitz and Stegun (1964), p. 378: for large z, π −z 4ν 2 − 1 e 1+ + O(z −2 ) . 2z 8z  Kν (z) ∼ Hence,  √ ψ(s; α) = 2Γ−1 (α)sα/2 Kα (2 s ) ∼ 2Γ−1 (α)sα/2  π 1/2 1/2 e−2s = π 1/2 Γ−1 (α)sα/2−1/4 e−2s , 1/2 4s  s → ∞. (3.28)  Also, as s → ∞, √ 1/2 ψ (1) (s; α) = −2Γ−1 (α)s(α−1)/2 Kα−1 (2 s ) ∼ −π 1/2 Γ−1 (α)sα/2−3/4 e−2s . For the d-dimensional Archimedean copula, then by Proposition 3.8 with a1 = π 1/2 Γ−1 (α), q = α/2 − 1/4, a2 = 2 and r = 1/2 in (3.10), as u → 0, ψ dψ −1 (u) ∼  √ dα/2−1/4 ζ d (− log u) u , ζ 2 aκ−1 1  a1 = π 1/2 Γ−1 (α), ζ = (α − 1/2)(1 − Thus κL (Cψ ) =  √  √ d ).  d.  Proof of Proposition 3.10: Suppose that K is a multivariate max-id copula such that, for any index set ∅ = I ⊂ Id , K I ((1 − s)1|I| ) ∼ saI I (s), s → 0+ , with 1 < aI and  I (s)  ∈ RV0 (0+ ). Note that (−1)|I| K I (1 − s)1|I| ,  K((1 − s)1d ) = 1 +  (3.29)  ∅=I⊂Id  where K I is the survival function of the I-margin copula KI and let K {i} (1−s) = s for any i ∈ Id . Letting s = 1 − exp{−ψ −1 (u)}, as u → 1− , i.e., s → 0+ , since aI > 1 for any ∅ = I ⊂ Id , 1 − ds dominates the right-hand side of (3.29), and  76  thus, − log K e−ψ  −1 (u)  1d = − log K((1 − s)1d ) ∼ − log (1 − ds) ∼ ds = d(1 − exp{−ψ −1 (u)}) ∼ dψ −1 (u).  Therefore, C(u1d ) ∼ ψ dψ −1 (u) = Cψ (u1d ),  u → 1− .  By Proposition 3.6, we know that C has intermediate upper tail dependence, and moreover, κU (C) = κU (Cψ ). This proves (a). To prove (b), note from Proposition 3.3 that κU (KI ) = 1 for any marginal copula KI . Assuming  I (s)  → hI ∈ (0, 1] as s → 0+ and h{i} = 1 for all i,  − log KI e−ψ  −1 (u)  where h∗I =  |J|−1 h . J ∅=J⊂I (−1)  1|I| = − log KI ((1 − s)1|I| ) ∼ − log (1 − h∗I s) ∼ h∗I ψ −1 (u), By the construction of (3.12), for any I-margin  copula of C,  CI (u1|I| ) = ψ − log KI (e−ψ  −1 (u)  1|I| ) ∼ ψ h∗I ψ −1 (u) .  Thus, as s → 0+ , i.e., u → 1− , (−1)|I| CI (u1|I| )  C(u1d ) = 1 + ∅=I⊂Id  (−1)|I| ψ h∗I ψ −1 (u) =  ∼1+ ∅=I⊂Id  (−1)|I|−1 1 − ψ h∗I ψ −1 (u) ∅=I⊂Id  If 1 − ψ(x) ∈ RVβ (0+ ), then clearly, (−1)|I|−1 (h∗I )β ,  C((1 − u)1d ) ∼ u ∅=I⊂Id  So, κU (C) = 1 and λU (C) =  |I|−1 (h∗ )β . ∅=I⊂Id (−1) I  77  u → 0+ .  .  Proof of Proposition 3.11: Suppose that a d-variate max-id copula K(s1d ) ∼ sα (s) as s → 0+ and let s = exp{−ψ −1 (u)}. As u → 0+ , thus s → 0+ , − log K e−ψ  −1 (u)  1d = − log K(s1d ) ∼ − log(sα (s)) ∼ −α log s = αψ −1 (u).  Therefore, C(u1d ) = ψ − log K e−ψ  −1 (u)  1d  ∼ ψ αψ −1 (u) ,  u → 0+ .  With some modification of the proof of Theorem 3.3 of Charpentier and Segers (2009), we can prove the rest. For purpose of notational convenience, we include the modification in the following. Letting Λ = (α + dν)/(1 + ν) and ω(s) = −ψ(s)/ψ (s), then we know that ψ −1 (sx) − ψ −1 (s) = − log(x), ω(ψ −1 (s))  lim  s→0+  and if y(t) → y ∈ R as t → ∞, then ψ(t + y(t)ω(t)) = exp(−y). t→∞ ψ(t) lim  For any t > 0, write ψ(αψ −1 (ut)) = ψ αψ −1 (u) + y(u, t)ω(αψ −1 (u)) , where y(u, t) =  α[ψ −1 (ut) − ψ −1 (u)] ω(ψ −1 (u))  ω(ψ −1 (u)) . ω(αψ −1 (u))  As u → 0+ , y(u, t) → −α log(t)α−β = −α1−β log(t). Therefore, lim  u→0+  ψ −1 (αψ −1 (ut)) 1−β = exp(α1−β log(t)) = tα , −1 −1 ψ (αψ (u))  and thus C(u1d ) ∈ RVα1−β (0+ ). We have also known from Theorem 3.3 of Charpentier and Segers (2009) that κL (Cψ ) = d1−β . This completes the proof.  78  Chapter 4  Tail comonotonicity 4.1  Introduction  Suppose we have bivariate loss data and hope to estimate some high-risk scenarios, say by VaR or CTE. A widely used method is to fit some parametric models based on the bivariate Student t or other parametric copula families. Then risk measures or tail dependence can be derived from these fitted models. Due to model uncertainty, all of these methods are not conservative from the viewpoint of an actuary. The middle part might influence our estimation more than the important tail part does. These traditional methods are too sensitive to the middle part which contains most of the data (Nikoloulopoulos et al., 2012). Given that we do not have enough information for the joint tail of losses, we conservatively assume that it is upper tail comonotonic (as defined in Section 4.2), and the conservativity can be justified in Chapter 5. In this way, we actually give up estimating the “first order tail parameter” (i.e., the usual tail dependence parameter λ) by assuming a conservative one (i.e., λ = 1), and let the likelihood of data contribute to the estimation of the “second order tail parameter”. Since the first order parameter is only for an asymptotic property, the conservative assumption on it would not put too much constraint on the model. This approach will give us a more robust method of measuring risks. In the literature on actuarial science and quantitative finance, many efforts have been done to seek finer upper bounds for dependence structures. The concepts of 79  comonotonicity, conditional comonotonicity, and most recently, the upper comonotonicity have been studied to provide theoretically tractable bounds. However, these conditions are either too strong or not tailored for the tail, and could lead to an over-conservative risk measure. Moreover, those dependence structures lack flexible distribution families that can be used to model real data. We refer to Dhaene et al. (2002a,b); Cheung (2007, 2009) for the reference of these concepts. Tail comonotonicity, on the other hand, needs a weaker condition and requires that the degree of positive dependence approaches its maximum only when all the marginal losses go to infinity. The degree of dependence at the sub-extremal level can still be estimated from the data. This approach should better balance the requirements of safety and accuracy for risk management. Tail comonotonicity is also referred to as asymptotic full dependence or a dependence structure where the tail dependence parameter satisfies λ = 1. Although such a dependence structure is not new, we find that it has interesting properties, such as asymptotic additivity of VaR and CTE, that are analogous to the usual comonotonicity. Moreover, some parametric families illustrate its suitability for modeling loss data that may appear to have tail dependence. Among many copula families, Archimedean copulas and copulas constructed from a scale mixture of a non-negative random vector can be used to provide a tail comonotonic dependence structure. This chapter is organized as follows. In Section 4.2, the concept of tail comonotonicity, its properties and some parametric examples will be studied. Methods of constructing copulas with tail comonotonicity or the strongest tail dependence are given in Section 4.3. Asymptotic additivity of VaR and CTE under the assumption of tail comonotonicity is shown in Section 4.4. Finally, in Section 4.5, we conclude the chapter and propose some directions for future research.  4.2  Definitions of tail comonotonicity and properties  Let X = (X1 , . . . , Xd )T be a non-negative random vector, representing amounts of d losses, and the univariate marginal cdf’s are all continuous and denoted as F1 , . . . , Fd . The Fr´echet space containing all the random vectors with these univariate margins is denoted as R(F1 , . . . , Fd ). For risk management, the joint tail  80  probability P[X1 > x1 , . . . , Xd > xd ] is relevant, especially when x1 , . . . , xd are large values. Cheung (2009) studied an upper comonotonicity structure: roughly speaking, beyond a finite threshold a∗ , the dependence structure becomes comonotonicity, while keeping the dependence structure below the threshold flexible; this means that the distribution is not absolutely continuous. Using the Florida flooding dataset mentioned in Figure 1.2, the idea of comonotonicity and upper comonotonicity can be illustrated by Figure 4.1. Figure 4.1: Florida flood data - Comonotonicity vs Upper comonotonicity  ● ●  2  1  0  −1  −2 ●  ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ●  ●  ● ●● ●● ●●  ● ●  −2  −1  0  1  Florida flood aggregate loss (1977~2006) Normal score of aggregate loss of other 30 counties  Normal score of aggregate loss of other 30 counties  Florida flood aggregate loss (1977~2006)  2  ● ●  2  1  ●  0  ●  −1 ●  −2  ● ● ● ● ●● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ●● ● ● ● ● ●● ●●●● ●● ● ● ● ● ● ● ●● ●● ● ●● ●● ●● ● ●●●● ● ● ●●● ●● ●● ● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ●●●● ●● ●● ●● ● ● ●● ●● ● ● ● ● ●● ●●● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ●●● ●● ●● ● ● ● ● ●●●● ● ●● ● ● ● ● ● ● ●●●●● ●● ● ●● ● ● ●● ●● ● ●●● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●● ● ● ● ●● ● ● ●● ●● ● ● ●● ● ●● ● ●● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●●  ●  −2  Normal score of aggregate loss of 30 counties  ●  ● ●● ●● ●●  −1  0  1  2  Normal score of aggregate loss of 30 counties  Under this dependence structure, if x ≥ a∗ , then clearly P[X1 > x1 , . . . , Xd > xd ] = min{F 1 (x1 ), . . . , F d (xd )} = 1 − max{F1 (x1 ), . . . , Fd (xd )}, which is the upper bound for such a tail probability of any X ∈ R(F1 , . . . , Fd ). The upper bound coincides with the tail probability for the usual comonotonicity. Letting C be the survival copula of X, P[X1 > x1 , . . . , Xd > xd ] = C(F 1 (x1 ), . . . , F d (xd )). A proper threshold a∗ might not exist in real applications. However, note that our aim of proposing a conservative dependence structure will be met if the tail 81  probability can be well approximated by the upper bound when x is sufficiently large; that is, C(F 1 (x1 ), . . . , F d (xd )) ≈ min{F 1 (x1 ), . . . , F d (xd )},  xi ’s are sufficiently large.  When x is sufficiently large, F i (xi )’s become sufficiently close to 0. But how xi ’s converge to the right end points of their supports will affect the joint tail probability. Considering that the usual comonotonicity is a copula property and does not depend on the margins, here we want to define a dependence concept which will not rely on margins as well. Therefore, letting ui := F i (xi ), it suffices to have that C(u1 , . . . , ud ) ≈ min{u1 , . . . , ud },  ui ’s are sufficiently small.  To be convenient in real applications, this condition should also be satisfied with copulas C that are absolutely continuous. Of course, how ui ’s approach 0 will affect the above approximation. However, as long as the rates of convergence of ui ’s to 0 are comparable (in the sense that ui = uwi for any given 0 < wi < +∞), the above approximation should be good. Then C would be what we want if it satisfies C(uw1 , . . . , uwd ) ∼ min{uw1 , . . . , uwd },  u → 0+ ,  wi , . . . , wd ∈ [0, +∞),  i.e., lim  u→0+  C(uw1 , . . . , uwd ) = min{w1 , . . . , wd }, u  wi , . . . , wd ∈ [0, +∞), (4.1)  where the trivial case of wi = 0 for some i is also included. Definition 4.1 A random vector X is said to be upper tail comonotonic if X has a copula C and its survival copula C satisfies (4.1); the copula C is said to be an upper tail comonotonic copula. X is said to be lower tail comonotonic if X has a copula C that satisfies limu→0+ C(uw1 , . . . , uwd )/u = min{w1 , . . . , wd }, wi ∈ [0, +∞); the copula C is said to be a lower tail comonotonic copula.  82  Remark 4.1 Based on the above definition, tail comonotonicity is a concept about copula and does not rely on the marginal distributions as long as the conditions hold. In Definition 4.1, the random vector X is not necessarily continuous and the right end points of each univariate margin can be finite or infinite. Remark 4.2 In the framework of multivariate regular variation (MRV), the concept of upper tail comonotonicity is represented by the limit Radon measure (Resnick 2007 Resnick (2007)) ν([0, x]c ) = (min{x1 , . . . , xd })−α . So asymptotic full dependence for MRV is a case of upper tail comonotonicity with the univariate margins being power laws. Recall from Joe et al. (2010) that, when C is a d-variate copula with the survival function C, then lower tail dependence means that there exists a non-zero homogeneous function b of order 1 such that lim C(uw1 , . . . , uwd )/u = b(w1 , . . . , wd ),  u→0+  wi ∈ [0, +∞),  (4.2)  and upper tail dependence means that there exists a non-zero homogeneous function b∗ of order 1 such that lim C(1 − uw1 , . . . , 1 − uwd )/u = b∗ (w1 , . . . , wd ),  u→0+  wi ∈ [0, +∞). (4.3)  So, tail comonotonicity simply means that the lower and/or upper tail dependence functions are min{w1 , . . . , wd }. Also, note that for any tail dependence function b(w), we must have b(w) ≤ min(w1 , . . . , wd ). Some relevant properties for upper/lower tail comonotonicity are mentioned in Beirlant et al. (2004), we give some alternative results in the following two propositions. Proposition 4.1 Suppose C is a copula. Then λU (C) = 1 if and only if the upper tail dependence function exists and b∗ (w1 , . . . , wd ) = min(w1 , . . . , wd ). In 83  parallel, λL (C) = 1 if and only if the lower tail dependence function exists and b(w1 , . . . , wd ) = min(w1 , . . . , wd ). Remark 4.3 From Proposition 4.1, we can conclude that the concept of upper/lower tail comonotonicity is nothing but the tail dependence parameter λ = 1. The designation of this concept is just consistent with the names of upper/lower tail dependence. If the tail dependence function exists, then it must have the unique form. It is well known that all pairs of a random vector are comonotonic is equivalent to that the random vector is comonotonic. A parallel result also holds for upper/lower tail comonotonicity. Proposition 4.2 Suppose C is a d-variate copula, then any bivariate marginal copula C{ij} , i = j, is upper [resp. lower] tail comonotonic if and only if C is upper [resp. lower] tail comonotonic. Next, we give two examples of parametric tail comonotonic copulas that are two-parameter Archimedean copulas. Example 4.1 (BB2 in Joe and Hu (1996), Joe (1997)) With LT ψ(s) = [1 + δ −1 log(1 + s)]−1/θ , the bivariate Archimedean copula is −θ −1)  Cψ (u, v) = 1 + δ −1 log eδ(u  + eδ(v  −θ −1)  −1  −1/θ  ,  θ > 0, δ > 0.  Then as u → 0+ , Cψ (uw1 , uw2 ) ∼ 1 + δ −1 max{δ((uw1 )−θ − 1), δ((uw2 )−θ − 1)} u = min(w1 , w2 ).  −1/θ  u−1  Thus Cψ is lower tail comonotonic. Scatter plots of the BB2 copula with parameters δ = 0.2 and θ = 0.4 or 0.2 are in Figure 4.2 (N = 2000); there is no upper tail dependence for this copula. 84  Figure 4.2: Simulation of BB2 with/without the univariate margins being transformed to the standard Normal; in the left and middle plots δ = 0.2, θ = 0.4 and in the right plot δ = 0.2, θ = 0.2 BB2 d= 0.2 th= 0.4 (Uniform margin)  BB2 d= 0.2 th= 0.4 (Normal margin)  ●● ●●● ● ● ● ●●●● ● ● ●● ● ● ●●● ●● ●● ● ● ●● ● ●●● ●● ●● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ●● ●● ●●● ●●● ● ●● ●●● ●● ● ● ●● ● ●●● ● ● ●● ●● ●●● ● ●●● ● ● ● ●● ●● ● ● ● ●●●● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ●● ●●●● ● ●●●● ●● ● ●● ●●● ● ● ● ● ●● ●● ●● ● ●●● ●● ●●● ● ●●● ● ● ● ●● ● ● ● ● ● ●● ●●●●●●● ●● ● ● ●●● ● ● ● ● ●● ●● ● ●●● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ●● ● ● ● ● ●●● ●●● ● ● ● ● ● ●● ●● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ● ●● ●● ● ●● ● ● ● ●● ●● ● ● ● ●● ● ●● ● ● ● ● ● ●● ●● ● ●● ● ●● ● ●● ● ● ●● ●● ● ● ●●● ●● ●● ● ● ● ● ● ● ● ●●●● ●● ● ● ● ● 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Then as u → 0+ , Cψ (uw1 , uw2 ) θ ∼ exp −[δ −1 log(eδ(− log[u min(w1 ,w2 )]) )]1/θ u−1 = min(w1 , w2 ). u Thus Cψ is also lower tail comonotonic. Scatter plots of the BB3 copula with parameters δ = 0.2 and θ = 1.7 or 1.3 are in Figure 4.3 (N = 2000); there is also upper tail dependence for this copula and λU = 2 − 21/θ .  Remark 4.4 Looking at the plots for the lower comonotonic copulas, they appear suitable as survival copulas to be used to get conservative dependence structure for joint large losses. Although they are both lower tail comonotonic, there is not much constraint on the sub-extremal level.  85  Figure 4.3: Simulation of BB3 with/without the univariate margins being transformed to the standard Normal; in the left and middle plots δ = 0.2, θ = 1.7 and in the right plot δ = 0.2, θ = 1.3 BB3 d= 0.2 th= 1.7 (Uniform margin)  BB3 d= 0.2 th= 1.7 (Normal margin)  ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ●● ● ● ● ● ● ● ●●● ● ● ● ●● ●● ● ● ●● ●● ● ● ● ●● ● ● ●● ● ● ● ●● ●●● ● ● ● ● ● ●● ●● ●● ● ● ● ●● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ●● ●● ● ●● ● ● ●● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ●●● ● ● ●●● ● ● ● ●● ● ● ●● ●● ● ● ● ●● ● ●● ● ● ● ●● ● ●●● ● ● ●● ● ● ● ● ●●●●●● ●● ● ●● ● ● ●● ● ● ●● ● ●● ● ●● ● ● ●●● ●●●●● ● ● ● ● ●● ● ● ●● ● ●● ●● ● ● ● ● ●● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ●●● ● ●● ●●● ● ● ● ● ●● ● ● ●● ●● ● ●● ● ●● ● 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of tail comonotonic copulas  In this section, we propose methods to construct tail comonotonic copulas based on mixing distributions with very heavy tails; “very heavy” means there are no moments of any positive/negative orders or the corresponding survival function is slowly varying. Precise conditions will be presented in the following subsections.  4.3.1  Archimedean copulas  In this subsection, we will relate the tail behavior of LTs to upper/lower tail comonotonicity of corresponding Archimedean copulas. The next result says that if the LT ψ is slowly varying at the tails, then Cψ must have the tail dependence function min(w1 , . . . , wd ). Proposition 4.3 Let the Archimedean copula Cψ be based on the LT ψ satisfying ψ(∞) = 0. If ψ(s) ∈ RV0 then the lower tail dependence function exists and Cψ is lower tail comonotonic; if 1 − ψ(s) ∈ RV0 (0+ ) then the upper tail dependence function exists and Cψ is upper tail comonotonic. Remark 4.5 Write the Archimedean copula as the mixture ∞ d  Gη (ui )dFH (η),  Cψ (u1 , . . . , ud ) = 0  86  i=1  where G(u) = exp{−ψ −1 (u)} is a cdf on [0, 1] and H is a resilience random variable with LT ψ. The condition 1 − ψ ∈ RV0 (0+ ) means that the density fH (η) has a very heavy tail as η → ∞ so that E[H m ] = ∞ for all m > 0. The condition ψ ∈ RV0 means that the density fH (η) has a very heavy tail as η → 0 so that E[H −m ] = ∞ for all m > 0. The implication of the condition of 1−ψ at 0 follows by the proof of Lemma 3.4. Similarly, we can show the implication of the condition of ψ at ∞ as follows. Suppose to the contrary that E[H −m ] < ∞ for some m > 0. Let Wm = (Z/H)m , where Z ∼ Exponential(1), independent of H, and H has the LT ψ. Then P[Wm ≥ w] = P[Z ≥ Hw1/m ] = ψ(w1/m ).  Then  E[H −m ]  ∞ 1/m )F (dy) H 0 exp(−yw  =  < ∞ and Z having all positive moments implies that  E[Wm ] < ∞ and thus wP[Wm ≥ w] → 0, as w → ∞; that is, wψ(w1/m ) → 0, or equivalently sm ψ(s) → 0, as s → ∞. It is well known that if ψ(s) ∈ RV0 and m > 0, we must have sm ψ(s) → ∞. The contradiction implies that E[H −m ] = ∞ for all m > 0. It can be verified that the LTs for BB2 and BB3 are both slowly varying at +∞, except in the lower boundary case (i.e., θ = 1 for BB3); that is, ψBB2 (s) = [1 + δ −1 log(1 + s)]−1/θ ∈ RV0 , ψBB3 (s) = exp −[δ −1 log(1 + s)]1/θ  θ > 0, δ > 0;  ∈ RV0 ,  θ > 1, δ > 0.  Now, consider the tail behavior of the LT studied in (3.10); that is, assume that a LT η satisfies η(s) ∼ T (s) = a1 sq exp{−a2 sr } and η (s) ∼ T (s),  s → ∞, with a1 > 0, a2 ≥ 0,  (4.4)  where r = 0 implies a2 = 0 and q < 0, and r > 0 implies r ≤ 1 and q can be 0, negative or positive. Note that r > 1 or r < 0 is not possible because of the complete monotonicity property of a LT. This condition covers almost all of the LT families in the Appendix of Joe (1997), as well as other LT families that can be obtained by integration or differentiation. The next result is contained in the Appendix of Joe (1997), where L∗+∞ is the class of infinitely differentiable increasing functions of [0, ∞) onto [0, ∞), with 87  alternating signs for derivatives. This combines some results from pages 441 and 450 of Feller (1971); the condition on φ implies that it is the LT of an infinitely divisible random variable. Lemma 4.4 If φ is a LT such that − log φ ∈ L∗+∞ and η is another LT, then ψ(s) = η(− log φ(s)) is a LT. Proposition 4.5 Suppose the LT η(t) satisfies condition (4.4), and take LT of Gamma φ(s) = (1 + s)−1/θ , thus ψ(s) := η(− log(φ(s))) is also a LT. Then 0 ≤ r < 1 implies that ψ(s) ∈ RV0 . In the literature, risks are compared with respect to some stochastic orders, say, the usual stochastic order and the increasing convex order (M¨uller and Stoyan, 2002). In our opinion, the comparisons of risks are more meaningful for high-risk scenarios. However, the conditions these stochastic orders need to satisfy are too strong to be flexible for comparing tail risks. To this end, we may define some new concepts of stochastic order that are particularly used to compare the tails of univariate/multivariate cdf’s. For example, we may define a stochastic order based on asymptotic properties of distribution functions as what follows. Definition 4.2 (Ultimate usual stochastic orders) Let X, Y be random variables, X is said to be greater than Y in the upper ultimate usual stochastic order if there exists a finite q such that F X (t) ≥ F Y (t) for all t ≥ q, written as X  st  Y . Let X  and Y be random vectors, then X is said to be greater than Y in the upper ultimate usual stochastic order if there exists a finite threshold q such that F X (t) ≥ F Y (t) for all t ≥ q, written as X notation is  st  st  Y. For lower ultimate usual stochastic order, the  . Then X is said to be greater than Y in the lower ultimate usual  stochastic order (written as X st Y ) if there exists a finite q such that FX (t) ≥ FY (t) for all t ≤ q, and X is said to be greater than Y in the lower ultimate usual stochastic order if there exists a finite threshold q such that FX (t) ≥ FY (t) for all t ≤ q, written as Xst Y. The relationships can also be symbolized by using corresponding cdfs, such as, FX st FY for Xst Y. If X, Y represent amounts of losses, then X  st  Y implies that X is riskier  than Y . If the right tail of X is heavier than that of Y , then this is a sufficient condition for X  st  Y. 88  Proposition 4.6 Let ψ1 , ψ2 be given LTs. If ψ1−1 ◦ ψ2 (s) is superadditive for sufficiently large s, then there exists a finite threshold q such that for any u ≤ q, Cψ1 (u) ≥ Cψ2 (u), i.e., Cψ1 st Cψ2 . Note: using notation of Archimedean copula in Nelsen (2006), the generator should be strict in order to apply Proposition 4.6. Remark 4.6 From Proposition 4.6, we expect that BB2 is more lower tail positive dependent than BB3 below a sufficiently low threshold, although they are both lower tail comonotonic. Let ψ1 (s) := ψBB2 (s) = [1 + δ1−1 log(1 + s)]−1/θ1 and ψ2 (s) := ψBB3 (s) = exp −[δ2−1 log(1 + s)]1/θ2 , then 1/θ2  g(s) := ψ1−1 ◦ ψ2 (s) = exp{δ1 (eθ1 (log(1+s)/δ2 )  − 1)} − 1.  Letting h(s) := log(1 + s)/δ2 , 1/θ2  g (s) = δ1 θ1 h(s)1/θ2 eθ1 h(s)  1/θ2  e  δ1 eθ1 h(s)  −1  /(θ2 (1 + s) log(1 + s)).  By observing g (s), we know that g(s) is strictly increasing and ultimately strictly convex. Assume that g(s) is strictly convex as s ≥ s0 , and let y0 := g(s0 ) and z(s) := g(s + s0 ) − y0 , then z(s) is superadditive for s ∈ [0, ∞). Therefore, for x, y ≥ 0, z(x + y) ≥ z(x) + z(y); that is, g(x + y + s0 ) + g(s0 ) ≥ g(x + s0 ) + g(y + s0 ). Since g(s) is strictly increasing and strictly convex as s ≥ s0 , we must have, when x, y are sufficiently large, g(x + y + 2s0 ) ≥ g(x + y + s0 ) + g(s0 ). Therefore, ψ1−1 ◦ ψ2 (s) is ultimately superadditive.  4.3.2  Heavy tail mixtures  Consider the heavy tail scale mixture X = (RT1 , . . . , RTd ),  (4.5)  where R and Ti ’s are all non-negative random variables, R is independent of the Ti ’s, and the dependence structure between Ti ’s is not specified. This form covers truncated elliptical distributions. The CTE for such random vectors has been 89  studied in Zhu and Li (2012), and more relevant study is in Chapter 6, where R is assumed to be regularly varying and second order regularly varying in the right tail, respectively. The following lemma is often referred to as the Breiman’s Theorem (Breiman (1965)). Although the proof in Breiman (1965) is not for a general α, it can be adapted for proving a more general case where 0 ≤ α < +∞. To the best of our knowledge, we have not found a complete and detailed proof for this case. So we include a proof in Section 4.6, and emphasize that the result for α = 0 corresponding to slowly variation of Y also holds. Lemma 4.7 Suppose a random variable R ≥ 0 with F R (y) ∈ RV−α , (0 ≤ α < +∞). T ≥ 0 is independent of R and E[T α+δ ] < ∞ for some δ > 0. Then lim  y→∞  P[RT > y] = E[T α ]. P[R > y]  (4.6)  Proposition 4.8 Let X = (RT1 , . . . , RTd ) be defined as in (4.5). If R ∈ RV0 , E[Tiδi ] < ∞, i = 1, . . . , d, for some δi > 0, then X is upper tail comonotonic. The above scale mixture can be used to construct upper tail comonotonic copulas. Note that, the margins Ti ’s are not necessarily identical and their dependence structure is not specified. An elliptical random vector X can be written as d  X = µ + RAU ∼ Ed (µ, Σ, φ),  (4.7)  where the radial random variable R ≥ 0 is independent of U, U is an k-dimensional random vector uniformly distributed on the surface of the unit hypersphere S2k−1 = {z ∈ Rk |zT z = 1}, A is a d × k matrix with rank(A) = k and AAT = Σ that is positive semidefinite, φ is the characteristic generator. When Σ is positive definite, write ρij := Σij /  Σii Σjj , and ρij = ±1 in the following to avoid trivial  cases. We refer to Fang et al. (1990) for a comprehensive reference for elliptical distributions. Now, we are studying whether there exist tail comonotonic elliptical distributions. Note that the Ti ’s in Proposition 4.8 are all non-negative, so it does not cover 90  the usual elliptical distributions. For tail dependence of elliptical distributions, Schmidt (2002), and Hult and Lindskog (2002) study the case where the radial random variable R has certain tail patterns such as regularly varying right tails. Frahm et al. (2003) shows that the two different representations of tail dependence parameters in Schmidt (2002) and Hult and Lindskog (2002) for the regularly varying case are equivalent. From their results, we know that, with a regularly varying radial random variable R (tail index −α < 0), the non-degenerate elliptical distributions have tail dependence parameters that are strictly less than 1. We now show that, even when P[R > s] is slowly varying in s, tail comonotonicity still does not hold. The intuitive reasoning for this result is that for bivariate elliptical distributions that have usual tail dependence, there is tail dependence in all corners/quadrants, so that the tail dependence in the upper positive quadrant is not 1. Proposition 4.9 Suppose X is an elliptical random vector as defined in (4.7) with Σii > 0 for i = 1, . . . , d, if P[R > s] ∈ RV0 , then any bivariate margins of X is upper (and lower) tail dependent, and the tail dependence parameter λij = 1/2 + (1/π) arcsin ρij . Remark 4.7 In the proof of Theorem 4.3 in Hult and Lindskog (2002), it is claimed d  that X = RAU implies that  Xi Xj  d  = R  √  Σii    0  Σjj ρjj  Σjj  cos ϕ 1−  ρ2ij    sin ϕ  ,  where ϕ ∼ Uniform(−π, π). Here we need to make a note that the above claim is true only when X is bivariate. For a general d-dimensional X (d > 2), radial random variables for margins will not be the same as the original R for the whole distribution. We make a more detailed argument about this in what follows. Due to the Corollary in page 43 of Fang et al. (1990), margins of elliptical distributions are still elliptical with the same characteristic generator φ. Moreover, due to Lemma 5.3 of Schmidt (2002), the radial random variable R depends on the dimension d of the elliptical distribution, and there exists a constant k > 0 such  91  that √ d Rs = kRd B,  B ∼ Beta(s/2, (d − s)/2),  s < d,  and B is independent of Rd . Since a Beta distribution has moments of all orders, by Lemma 4.7, Rs inherits the tail behavior of Rd when Rd is regularly varying or slowly varying, which is why the original proof in Hult and Lindskog (2002) can still yield a correct conclusion. Let (U1 , U2 ) follows a bivariate copula, where Ui ∼ Uniform(0, 1), i = 1, 2. From Embrechts et al. (2009a), the tail dependence parameters in the “North East (NE)” and “South East (SE)” for the bivariate Student t copula are the following:  λNE := lim P[U2 > u, |U1 > u] = u→1−  π/2 ν (π/2−arcsin ρ)/2 (cos t) dt π/2 (cos t)ν dt 0  =: Λ(ν, ρ); (4.8)  λSE := lim P[U2 ≤ 1 − u, |U1 > u] = u→1−  π/2 ν (π/2+arcsin ρ)/2 (cos t) dt π/2 (cos t)ν dt 0  =: Λ(ν, −ρ). (4.9)  Note that λNE + λSE is still a conditional probability and thus 0 ≤ λNE + λSE ≤ 1; in fact, for the above 0 < Λ(ν, ρ) + Λ(ν, −ρ) < 1 for ν > 0. Moreover, both λNE and λSE are decreasing in ν: for a bivariate tν copula (Embrechts et al. (2002), Joe (2011)), λNE = 2Tν+1 − (ν + 1)(1 − ρ)/(1 + ρ) ;  (4.10)  λSE = 2Tν+1 − (ν + 1)(1 + ρ)/(1 − ρ) ,  (4.11)  and it can be verified that (4.8) and (4.10) are equivalent, and (4.9) and (4.11) are equivalent (see Section 4.6 for a direct derivation). Since Tν+1 (x) is decreasing in ν for a fixed x ≤ 0, and Tν+1 (x) is increasing in x for a fixed ν, both λNE and λSE are decreasing in ν. From Theorem 5.2 of Schmidt (2002), we know that if there exists a bivariate  92  tail dependent margin, then we must have 0 < lim inf x→∞  F (tx) F (tx) ≤ lim sup ≤ 1, F (x) x→∞ F (x)  t ≥ 1,  where F (·) is the survival function of the radial random variable R. If R is slowly varying, then the upper bound in the above necessary condition is reached. Although we have not proved it for now, we conjecture that there are no tail comonotonic non-degenerate elliptical distributions and the maximum tail dependence parameter for a bivariate elliptical copula is λ = 1/2 + (1/π) arcsin ρ. As ν → 0+ in (4.8) and (4.9), the sum converges to [1/2 + (1/π) arcsin ρ] + [1/2 + (1/π) arcsin(−ρ)] = 1 and this is the maximum possible value of limx→∞ P(|X2 | > x|X1 > x) when X1 , X2 have a common distribution. Hence when all corners are considered, elliptical distributions with a slowly variable radial random variable have the strongest possible tail dependence. In contrast, note that Archimedean copulas based on LTs do not have tail dependence on quadrants that have different signs.  4.3.3  Extreme value copulas  We now show that for extreme value copulas, tail comonotonicity is equivalent to comonotonicity. Proposition 4.10 Suppose C is an extreme value copula, then λU = 1 ⇐⇒ C is a comonotonic copula ⇐⇒ λL = 1.  4.4  Asymptotic additivity of risk measures  As a dependence structure, tail comonotonicity may affect the risk measures of aggregated losses. In this section, we mainly study impacts of tail comonotonicity on additivity of commonly used risk measure such as VaR and CTE. It is well known that VaR and CTE are additive when the loss random variables are comonotonic (see Dhaene et al. (2006)); that is, if (X1 , . . . , Xd ) is comono-  93  tonic, then for all p ∈ (0, 1), d  d  Xi  VaRp i=1  =  d  VaRp (Xi ); i=1  d  Xi  CTEp i=1  =  CTEp (Xi ). i=1  The additivity property also holds for the upper comonotonicity in the sense of Cheung (2009) when the probability level p associated with the risk measures is larger than a threshold specified by the upper comonotonicity structure. A natural question is whether such an additivity property can be kept asymptotically as p → 1− . Asymptotic super and/or sub additivity of risk measures has been studied explicitly or implicitly in several papers. For example, Embrechts et al. (2009b) studied asymptotic additivity properties of VaR for multivariate (d > 2) dependent loss random variables that have regularly varying survival functions (with index −β) and Archimedean dependence structures. It is shown that for a probability level p < 1 and sufficiently close to 1, whether strict super or sub additivity depends on whether β < 1 or β > 1. The Archimedean copula has upper tail dependence but the upper tail dependence parameter λ < 1 (since the generator of the Archimedean copula is assumed to be regularly varying). In Section 3.3 of Alink et al. (2007), an example of lower tail comonotonic copula has been shown together with Corollary 2.4 of Embrechts et al. (2009b) that VaR is asymptotically additive for exchangeable regularly varying dependent random variables that have an upper tail comonotonic copula. Analogous to the additivity property of VaR and CTE for a comonotonic random vector, we find that, asymptotic additivity of VaR and CTE still holds for a large class of random vectors that are relevant for quantitative risk management. We now prove some results for asymptotic additivity of VaR and CTE for random variables that are in the maximum domain of attraction (MDA) of Fr´echet and Gumbel, respectively. The case for MDA of Weibull corresponds to loss random variables that are bounded above, and is not very relevant for actuarial applications, so we do not consider it here. Only upper tails of losses are to be studied as they are more relevant for risk measures; analogous results for lower tails also hold but are omitted here. It suffices  94  to consider non-negative random variables to study upper tails, so in the following the random vector X is assumed to be non-negative. However, for lower tails, we have simpler notation for tail comonotonicity. Therefore, instead of studying X directly, in what follows, we always define Y := −X and prove the results based on Y. The following corresponding assumptions are also assigned on Y, which have corresponding natural meanings for the non-negative random vector X. For study of tail behavior of random variables, we always assume that the distribution function is continuous (or at least continuous in the tail regions). So P[X < x] coincides with P[X ≤ x] in what follows.  Assumption 4.1 Let Y be a non-positive continuous random vector with marginal distributions F1 , . . . , Fd defined on (−∞, 0] such that Fi (−t) ∈ RV−α with α > 0, and lim Fi (−t)/F1 (−t) = ki ,  t→∞  0 < ki < ∞,  for any i = 1, . . . , d. Remark 4.8 If Y := −X satisfies Assumption 4.1, thus P[Xi > t] ∈ RV−α , then each univariate margin of X is in the MDA of Fr´echet (Theorem 3.3.7 of Embrechts et al. (1997)).  The following lemmas are analogous to Lemma 6.1 of Resnick (2007). They d  are useful in proving Propositions 4.13 and 4.14. For E ⊂ R , denote M+ (E) := {µ : Radon, non-negative measures on the Borel σ-algebra E of E}, and let the lower boundary LB := {(y1 , . . . , yd ) : some yi = 0} and the upper boundary U B := {(y1 , . . . , yd ) : some yi = ∞}, then define E1 := [0, ∞] \ LB; E2 := [−∞, ∞] \ U B.  95  Lemma 4.11 Suppose µn , µ ∈ M+ (E1 ). Then, as n → ∞ v  µn → µ in M+ (E1 ) ⇐⇒ µn ([y, ∞]) → µ([y, ∞]) for any y ∈ [0, ∞) \ LB such that µ(∂[y, ∞]) = 0. Remark 4.9 The set E1 excludes the lower boundary, since otherwise rectangles of the form [y, ∞] can not determine the vague convergence. If µ puts mass on axes such as the usual normalization for asymptotic independence in Section 6.5.1 of Resnick (2007), then µ ∈ / M+ (E1 ). The condition µ(∂[y, ∞]) = 0 also requires that µ does not put mass on the upper boundary. Lemma 4.12 Suppose µn , µ ∈ M+ (E2 ). Then, as n → ∞ v  µn → µ in M+ (E2 ) ⇐⇒ µn ([−∞, y]) → µ([−∞, y]) for any y ∈ (−∞, ∞] \ U B such that µ(∂[−∞, y]) = 0. Proposition 4.13 (Asymptotic additivity of VaR: Fr´echet case) Suppose X is nonnegative and upper tail comonotonic, and Y := −X satisfies Assumption 4.1. If S = X1 + · · · + Xd , then d  VaRp (S) ∼  VaRp (Xi ),  p → 1− .  i=1  Assumption 4.2 Let Y be a non-positive continuous random vector with marginal distributions F1 , . . . , Fd defined on (−∞, 0] and there exists a positive measurable function a(·) such that Fi (−t + a(t)s) = es , t→∞ Fi (−t) lim  ∀s ∈ R,  (4.12)  and limt→∞ Fi (−t)/F1 (−t) = ki with 0 < ki < ∞ for any i = 1, . . . , d. Remark 4.10 Note that the condition Fi (−t)/F1 (−t) → ki implies that we can take ai (t) = a(t) for all i without loss of generality, and a(t) = o(t), t → ∞. If Y := −X satisfies Assumption 4.2, then due to Theorem 3.3.27 of Embrechts 96  et al. (1997), each univariate margin of X is in the MDA of Gumbel; that is F Xi (t+ a(t)s)/F Xi (t) → e−s , s ∈ R. Remark 4.11 A mixture distribution can satisfy the tail equivalence condition F Xi (t)/F X1 (t) → ki . Without loss of generality, we can let 0 < ki ≤ 1. For example, let random variables X1 ∼ Exponential(1) and X2 =  with probability k2  Exponential(1),  with probability 1 − k2 .  0,  Then F X2 (x)/F X1 (x) → k2 . Proposition 4.14 (Asymptotic additivity of VaR: Gumbel case) Suppose X is nonnegative and upper tail comonotonic, and Y := −X satisfies Assumption 4.2. If S = X1 + · · · + Xd , then d  VaRp (S) ∼  VaRp (Xi ),  p → 1− .  i=1  The conditions that are studied for asymptotic additivity of VaR satisfy assumptions considered in Asimit et al. (2011), which investigates asymptotic proportionality between CTE and VaR. With the help of their results, we may conclude the asymptotic additivity of CTE in what follows. Proposition 4.15 (Asymptotic additivity of CTE: Fr´echet case) Suppose X is nonnegative and upper tail comonotonic, and Y := −X satisfies Assumption 4.1 with α > 1. If S = X1 + · · · + Xd , then d  CTEp (S) ∼  CTEp (Xi ),  p → 1− .  i=1  Proposition 4.16 (Asymptotic additivity of CTE: Gumbel case) Suppose X is nonnegative and upper tail comonotonic, and Y := −X satisfies Assumption 4.2. If  97  S = X1 + · · · + Xd , then d  CTEp (S) ∼  CTEp (Xi ),  p → 1− .  i=1  The above asymptotic additivity results suggest that when we use tail comonotonicity as the dependence structure for such marginal distributions, we should expect that the diversification benefit (see Definition 2.2) will decrease to 0 as p → 1− . However, the speed of decay is unknown by observing the asymptotic relationship only; that is, we do not know when p is sufficiently close 1 so that the additivity relationship of the risk measures is sufficiently good. The relevant study will involve a second order approximation, and it is out of the scope of the thesis.  4.5  Concluding remarks and future research  Tail comonotonicity, like comonotonicity and upper comonotonicity, provides a bound-like dependence structure, and it is more reasonable to be used to capture information from data. Tail comonotonicity has some parallel properties of the usual comonotonicity, such as asymptotic additivity of VaR and CTE. Among many different copula families, Archimedean copulas with a mixing distribution that has no moments of any positive orders (for upper tail comonotonicity) or no moments of any negative orders (for lower tail comonotonicity), and copulas based on scale mixtures with a slowly varying non-negative random variable can be used to construct tail comonotonic copulas. However, elliptical and extreme value copulas cannot provide useful tail comonotonic copulas. Since tail comonotonicity is only an asymptotic property, it may or may not provide a conservative dependence structure for sub-extremal levels of risks. In Chapter 5, we will study how conservative are risk measures, under the tail comonotonic dependence structure. Although asymptotic additivity properties of VaR and CTE we have proved has already covered a wide range of random vectors, an elegant proof for a most general case (if exists; note that additivity of VaR and CTE for the usual comonotonicity does not depend on margins) must be welcome. Many other topics, such as optimal portfolio design, could be studied under the assumption of tail comono98  tonicity. We think that tail comonotonicity is particularly useful for dealing with high dimensional data for high-risk scenarios. Due to the curse of dimensionality, it is very hard (or impossible) to accurately model the tail behavior of aggregate high dimensional risks. For dependence modeling of multivariate random variables, vine copula is promising (Bedford and Cooke, 2002; Aas et al., 2009; Joe et al., 2010). By Proposition 4.2, it is feasible to use bivariate tail comonotonic copulas to build up a vine copula that is still tail comonotonic. The vine copula with tail comonotonicity may provide a reasonable cushion for the overall dependence structure.  4.6  Proofs  Proof of Proposition 4.1: If we take (w1 , . . . , wd ) = (1, . . . , 1), then it is obvious that λ = 1. For the other direction, letting w∗ = min(w1 , . . . , wd ) = 0, then C(uw∗ , . . . , uw∗ ) C(uw1 , . . . , uwd ) min(uw1 , . . . , uwd ) ≤ ≤ = w∗ u u u C(uw∗ , . . . , uw∗ ) ∗ C(uw1 , . . . , uwd ) ⇒ w∗ = lim w ≤ lim inf ∗ uw u uw∗ →0+ u→0+ C(uw1 , . . . , uwd ) ≤ lim sup ≤ w∗ . u + u→0 Thus, b(w1 , . . . , wd ) = min(w1 , . . . , wd ). Similar for the upper tail. Proof of Proposition 4.2: By Proposition 4.1, it suffices to prove that λ = 1. We only state the proof for lower tail comonotonicity here. Suppose Ui ∼ Uniform(0, 1) with joint distribution C. Since P[U1 ≤ u] −  d i=2 P[U1  ≤ u < Ui ]  u  ≤  C(u1d ) ≤ 1, u  and pairwise lower tail comonotonicity implies that limu→0+  d i=2 P[U1  ≤u<  Ui ]/u = 0, which implies that C is lower tail comonotonic. The other direction is due to the fact that if λL (C) = 1, then λL (CI ) = 1 for every marginal copula CI , 1 < |I| < d. Proof of Proposition 4.3: By Proposition 4.1, we only need to prove the tail de99  pendence parameter λ = 1. For the lower tail, letting s := ψ −1 (u), then because ψ(s) ∈ RV0 , λL = lim  u→0+  Cψ (u1d ) ψ(dψ −1 (u)) ψ(ds) = lim = lim = 1. + s→+∞ u u ψ(s) u→0  For the upper tail, by Proposition 4.2, it suffices to prove the bivariate case. Let s := ψ −1 (1 − u), then λU = lim  u→0+  Cψ (u, u) ψ(2ψ −1 (1 − u)) − 1 = lim 2 + u u u→0+ 1 − ψ(2s) = 1, = lim 2 − + 1 − ψ(s) s→0  which finishes the proof for the upper tail. Proof of Proposition 4.5: Clearly, − log(φ(s)) = (1/θ) log(1 + s) ∈ L∗+∞ , by Lemma 4.4, ψ(s) is a LT. Then the conclusion is straightforward by plugging − log(φ(s)) into η(·) of (4.4) since exp(−[log(1 + s)]r ) ∈ RV0 as 0 < r < 1.  Proof of Proposition 4.6: Superadditivity of ψ1−1 ◦ ψ2 (s) for sufficiently large s implies that there exists sufficiently large s0 such that s1 ≥ s0 and s2 ≥ s0 lead to ψ1−1 ◦ ψ2 (s1 + s2 ) ≥ ψ1−1 ◦ ψ2 (s1 ) + ψ1−1 ◦ ψ2 (s2 ). Let s1 := ψ2−1 (u) and s2 := ψ2−1 (v), then u, v ≤ ψ2 (s0 ) implies that ψ1−1 ◦ ψ2 (ψ2−1 (u) + ψ2−1 (v)) ≥ ψ1−1 ◦ ψ2 (ψ2−1 (u)) + ψ1−1 ◦ ψ2 (ψ2−1 (v)); that is, ψ2 (ψ2−1 (u) + ψ2−1 (v)) ≤ ψ1 (ψ1−1 (u) + ψ1−1 (v)), which completes the proof. Proof of Lemma 4.7: Choose some b(y), such as y 0 (y) for a suitably chosen  100  slowly varying function  0 (y),  satisfying that as y → ∞, b(y) → ∞, and  bα+δ (y)F R (y) → ∞,  ∀δ > 0  (4.13)  b(y)/y → 0.  (4.14)  Then, P[RT > y] P[R > y] ∞ 0 F R (y/t)FT (dt) = lim y→∞ F R (y) lim  y→∞  = lim  b(y) F R (y/t)FT (dt) 0  F R (y)  y→∞ ∞  = lim  y→∞ 0  1(0,b(y)] (t)  + lim  ∞ b(y) F R (y/t)FT (dt)  y→∞  F R (y)  F R (y/t) FT (dt) + lim y→∞ F R (y)  ∞ b(y)  F R (y/t) FT (dt) F R (y)  =: lim I1 + lim I2 . y→∞  y→∞  We know 1(0,b(y)] (t) FFR (y/t) ≤ 1(0,1] (t) + 1(1,b(y)] (t) FFR (y/t) , and also, when t ∈ (y) (y) R  R  (1, b(y)], (4.14) implies that as y → ∞, y/t → ∞. Then as y is sufficiently large, by the Karamata’s representation of regularly varying function, when t ∈ (1, b(y)], F R (y/t) ≤ tα+δ . F R (y) Since E[T α+δ ] < ∞, by the dominated convergence theorem, ∞  lim I1 =  y→∞  lim 1(0,b(y)] (t)  0  y→∞  F R (y/t) FT (dt) = F R (y)  ∞  tα FT (dt) = E[T α ]. 0  For I2 , we have ∞  I2 ≤ b(y)  1 FT (dt) ≤ F R (y)  ∞ b(y)  tα+δ FT (dt) → 0, F R (y)bα+δ (y)  and the convergence to 0 is due to E[T α+δ ] < ∞ and (4.13).  101  y → ∞,  Proof of Proposition 4.8: Due to Lemma 4.7, F X1 (s) ∼ F Xi (s), s → +∞ for i = −1  −1  −1  2, . . . , d, thus, s = F Xi (F Xi (s)) ∼ F Xi (F X1 (s)). Since F X1 (t) = FX−11 (1 − t), clearly, FX−1i (FX1 (s)) ∼ s,  s → +∞;  i = 2, . . . , d.  (4.15)  Assuming F is the joint cdf of (T1 , . . . , Td ), and hi (s) := FX−1i (FX1 (s))/s, i = 1, . . . , d, the upper tail dependence parameter  λU = lim  t→1−  = lim  d  s→+∞  P X1 > FX−11 (t)  P RT1 > s, . . . , RTd > FX−1d (FX1 (s)) /P[R > s]  s→+∞  = lim  P X1 > FX−11 (t), . . . , Xd > FX−1d (t)  R+  P[RT1 > s]/P[R > s] P [R > s (max{h1 (s)/t1 , . . . , hd (s)/td })] F (dt1 , . . . , dtd )/P[R > s] .  P[RT1 > s]/P[R > s] (4.16)  Due to Lemma 4.7, lims→+∞ P[RT1 > s]/P[R > s] = 1. For the numerator of (4.16), choose some b(s) such as s 0 (s) with a proper slowly varying function  0  such that, s → ∞ implies b(s) → ∞, and b (s)F R (s) → ∞, b(s)/s → 0.  ∀ >0  (4.17) (4.18)  102  d  Denote (b(s)1d , +∞]c := R+ \ (b(s)1d , +∞], then lim  s→+∞ Rd +  = lim  s→+∞ Rd +  ×  P [R > s (max{h1 (s)/t1 , . . . , hd (s)/td })] F (dt1 , . . . , dtd ) P[R > s] 1(b(s)1d ,+∞]c (t1 , . . . , td )×  P [R > s (max{h1 (s)/t1 , . . . , hd (s)/td })] F (dt1 , . . . , dtd ) P[R > s]  + lim  s→+∞ Rd +  ×  1(b(s)1d ,+∞] (t1 , . . . , td )×  P [R > s (max{h1 (s)/t1 , . . . , hd (s)/td })] F (dt1 , . . . , dtd ) P[R > s]  =: lim I1 + lim I2 . s→+∞  s→+∞  Due to (4.15), for any 0 < γ < 1, there exists an s0 < ∞ such that, s > s0 implies that hi (s) > γ for all i. Then s > s0 implies that 1(b(s)1d ,+∞]c (t1 , . . . , td )  P [R > s (max{h1 (s)/t1 , . . . , hd (s)/td })] P[R > s]  ≤ 1[0,γ1d ) (t1 , . . . , td ) + 1(b(s)1d ,+∞]c \[0,γ1d ) (t1 , . . . , td )× ×  P [R > sγ (max{1/t1 , . . . , 1/td })] . P[R > s]  It is well known that P[R > s] is slowly varying implies P[R > st]/P[R > s] converges uniformly in t ∈ [a, b] as s → ∞, where 0 < a, b < ∞ (see Bingham et al., 1987). Then P[R > sγ(max{1/t1 , . . . , 1/td })]/P[R > s] converges uniformly in t ∈ (b(s)1d , +∞]c \ [0, γ1d ) to 1. Moreover, t ∈ (b(s)1d , +∞]c \ [0, γ1d ) and (4.18) implies that, as s → +∞, sγ max{1/t1 , . . . , 1/td } → +∞. Then slow  103  variation of P[R > s] and dominated convergence theorem implies that lim I1  s→+∞  =  lim 1(b(s)1d ,+∞]c (t1 , . . . , td )×  d R+ s→+∞  × =:  P [R > s (max{h1 (s)/t1 , . . . , hd (s)/td })] F (dt1 , . . . , dtd ) P[R > s] lim 1(b(s)1d ,+∞]c (t1 , . . . , td )Ω(s, t1 , . . . , td )F (dt1 , . . . , dtd ).  d R+ s→+∞  Since (4.15), for any 0 < plies that 1 −  1  1, 2  < 1, there exists an s1 such that s > s1 im-  ≤ hi (s) ≤ 1 +  2  for all i. Thus, for any t1 , . . . , td such that  min{t1 , . . . , td } > 0, P[R > s(1 + 2 )/ min{t1 , . . . , td }] ≤ lim sup Ω(s, t1 , . . . , td ) P[R > s] s1 <s→∞ P[R > s(1 − 1 )/ min{t1 , . . . , td }] ≤ lim = 1. s→∞ P[R > s]  1 = lim  s→∞  Therefore, lim I1 = 1.  s→∞  For I2 , P [R > s (max{h1 (s)/t1 , . . . , hd (s)/td })] P[R > s] 1 (min{t1 , . . . , td }) ≤ 1(b(s)1d ,+∞] (t1 , . . . , td ) ≤ 1(b(s)1d ,+∞] (t1 , . . . , td ) . P[R > s] P[R > s]b (s)  1(b(s)1d ,+∞] (t1 , . . . , td )  Note that, E[Tiδi ] < ∞ implies that E[(min{T1 , . . . , Td }) ] < ∞ for some > 0, because E[(min{T1 , . . . , Td })δi ] ≤  δi d t F (dt1 , . . . , dtd ) R+ i  = E[Tiδi ] < ∞. Since  E[(min{T1 , . . . , Td }) ] < ∞, and P[R > s]b (s) → +∞ (due to (4.17)), by dominated convergence theorem, lim I2 = 0.  s→+∞  Thus, the claim is proved. 104  Proof of Proposition 4.9: The proof in Theorem 4.3 of Hult and Lindskog (2002) remains valid for this case where R is slowly varying, although in their proof, R is required to be regularly varying with a tail index −α < 0. Proof of equivalence between (4.8) and (4.10): The cdf of tν+1 can be written as a regularized Beta function as follows: (Press, 2007, page 323) 1 1 1 ν+1 1 B x; ν+1 2 ,2 − , ν > 0; 0 < x := 2 − sign(t) < 1. ν+1 1 2 2 B 2 ,2 2 t +ν+1  Tν+1 (t) =  where B(x; ., .) and B(., .) are incomplete Beta and Beta functions. Using trigonometric function representation of Beta functions, we have with t = sin2 θ, B  ν+1 1 , 2 2 1  π/2  t(ν+1)/2−1 (1 − t)1/2−1 dt = 2  =  π/2  (sin θ)ν dθ = 2  0  0  1 Next, with B x; ν+1 2 , 2 , where t = −  (cos θ)ν dθ. 0  (ν+1)(1− ) 1+  and x =  ν+1 t2 +ν+1  = (1+ )/2,  then B  1+ ν+1 1 ; , 2 2 2 √ arcsin  (1+ )/2  =  s(v−1)/2 (1 − s)−1/2 ds  0  (1+ )/2  π/2  (sin θ)ν dθ = 2  =2 0  π/2−arcsin  √  (cos θ)ν dθ. (1+ )/2  Now using the identity cos(2θ) = cos2 θ − sin2 θ, it can be verified that 2(π/2 − arcsin  (1 + )/2) = π/2 − arcsin by taking “cos” on both sides and getting  . Therefore, the claim is proved. Proof of Proposition 4.10: By Proposition 4.2, it suffices to prove the bivariate case. For any multivariate extreme value copula C, there exists a function A : [0, ∞)d → [0, ∞) such that C(u1 , . . . , ud ) = exp{−A(− log u1 , . . . , − log ud )}, where A is convex, homogeneous of order 1 and satisfies that max{x1 , . . . , xd } ≤ A(x1 , . . . , xd ) ≤ x1 + · · · + xd . For a bivariate extreme value copula, the function x A(x, y) can be written as A(x, y) = (x + y)B( x+y ), where B(·) is convex and  105  max{w, 1 − w} ≤ B(w) ≤ 1 for 0 ≤ w ≤ 1 (see Joe, 1997, Theorem 6.4). The upper tail dependence parameter is λU = 2 − A(1, 1). If we let λU = 1, then B(1/2) = 1/2, also B(w) must be convex, so we have B(w) = max{w, 1 − w}; that is, A(x, y) = max{x, y} and thus C(u1 , u2 ) = exp{−A(− log u1 , − log u2 )} = min{u1 , u2 }. The other direction is straightforward. The lower tail order for a bivariate extreme value copula is κ = A(1, 1) (see Example 3.4). If λL = 1, we must have the lower tail order A(1, 1) = 1. Then the subsequent argument is the same as for the upper tail. Proof of Lemma 4.11: The proof is the same as Lemma 6.1 of Resnick (2007). We rewrite the proof here for completeness. (⇒) is due to Theorem 3.2 of Resnick + (E1 ), where (2007). For (⇐), let g ∈ CK + CK (E1 ) := {g : E1 → R+ , continuous with compact support},  then the support of g must be contained in some [y, ∞] such that µ(∂[y, ∞]) = 0. Since convergence on this set holds, supn µn ([y, ∞]) < ∞ and thus, sup µn (g) ≤ sup g(x) · sup µn ([y, ∞]) < ∞. n  n  x∈E1  + (E1 ), so {µn } is relatively compact due to (3.16) of This is true for any g ∈ CK  Resnick (2007). If µ and µ are two subsequential limits, then µ and µ agree on the continuity sets [y, ∞]. Additionally, the rectangles of those continuity sets [y, ∞] constitute the π-system which generates B(K o ) the Borel σ-algebra of K o , where K o := {B ⊂ E1 : B is relatively compact , µ(∂B) = 0}. Then µ = µ on E1 by Theorem 3.2 of Resnick (2007) again. Proof of Lemma 4.12: The proof here is similar to the proof of Lemma 4.11 by replacing [y, ∞] by [−∞, y]. Proof of Proposition 4.13: Let W = −S and Yi = −Xi , i = 1, . . . , d, then (Y1 , . . . , Yd ) is lower tail comonotonic, and the map y → P[Yi ≤ −y] ∈ RV−α . 106  Moreover, since limt→∞ Fi (−t)/F1 (−t) = ki for any i = 1, . . . , d, clearly, 1/α  Fi−1 (p) ∼ ki  F1−1 (p) as p → 0+ (Lemma 2.1 of Asimit et al. (2011), or Propo-  sition 0.8(vi) of Resnick (1987)). It suffices to prove that, d −1 FW (p)  ∼  1/α  F1−1 (p)  ki  p → 0+ .  ,  i=1  Also, P[W ≤ −t] ∈ RV−α (Proposition 7.3 of Resnick (2007)) and apply Proposition 0.8(vi) again. Then it suffices to show as t → ∞, α  d 1/α ki  P[W ≤ −t] ∼ P[Y1 ≤ −t]  .  (4.19)  i=1  Under Assumption 4.1, if Y is also lower tail comonotonic, and C is the copula, then we have for any y ∈ [0, ∞) \ LB, lim  t→∞  P[Y1 ≤ −ty1 , . . . , Yd ≤ −tyd ] P[Y1 ≤ −t] C  = lim  t→∞  F1 (−ty1 ) Fd (−tyd ) F1 (−t) F1 (−t), . . . , F1 (−t) F1 (−t)  F1 (−t)  = min{k1 y1−α , . . . , kd yd−α }, (4.20)  where the last equality is due to lower tail comonotonicity of C and uniform continuity of copula functions (Nelsen, 2006, see). Define a measure µ on E1 as µ([y1 , ∞] × · · · × [yd , ∞]) := min{k1 y1−α , . . . , kd yd−α }. This measure µ only puts mass on the line {(y1 , . . . , yd ) : k1 y1−α = · · · = kd yd−α }. If k1 y1−α = · · · = kd yd−α =: h, then µ([y1 , ∞] × · · · × [yd , ∞]) = h. Clearly, µ is Radon and every set [y, ∞] is a continuity set. Define µt (·) :=  P [(−Y1 /t, . . . , −Yd /t) ∈ ·] , P[Y1 ≤ −t]  107  then by Lemma 4.12 and (4.20), v  µt (·) → µ(·),  t → ∞.  v  where → is vague convergence. Define a set d  y ∈ [0, ∞) \ LB :  H :=  yi ≥ 1 . i=1  Then clearly, µ(∂H) = 0 and H is relatively compact. Thus by Theorem 3.2 of Resnick (2007), µt (H) → µ(H),  t → ∞.  That is, letting zi := ki yi−α , as t → ∞, P[W ≤ −t] = µt (H) → min k1 y1−α , . . . , kd yd−α : P[Y1 ≤ −t]  d  yi ≥ 1 i=1  d  = µLebesgue  z:  α  d  (z/ki )  −1/α  ≥1  1/α ki  =  i=1  ,  i=1  which justifies (4.19), thus finishing the proof. Proof of Proposition 4.14: Let W = −S and Yi = −Xi , i = 1, . . . , d, then (Y1 , . . . , Yd ) is lower tail comonotonic. Moreover, since lim Fi (−t + a(t)s)/F1 (−t) = ki es  t→∞  for any s ∈ R and i = 1, . . . , d, taking s = − log ki leads to VaRp (Yi ) ∼ VaRp (Y1 ) − a(−VaRp (Y1 )) log ki ,  108  p → 0+  (4.21)  Then it suffices to show as t → ∞, d  P W ≤ −dt − a(t)  log ki ∼ P[Y1 ≤ −t].  (4.22)  i=1  Under Assumption 4.2, if Y is also lower tail comonotonic, and C is the copula, then we have for any y ∈ (−∞, ∞] \ U B, lim  t→∞  P[Yi ≤ −t − a(t) log ki + a(t)yi , i = 1, . . . , d] P[Y1 ≤ −t] Fi (−t−a(t) log ki +a(t)yi ) F1 (−t)  C = lim  F1 (−t) y −log k i i ki e F1 (−t), i = 1, . . . , d  t→∞  = lim  · F1 (−t), i = 1, . . . , d  C  F1 (−t)  t→∞  = min{ey1 , . . . , eyd },  (4.23)  where the last equality is due to lower tail comonotonicity of C and uniform continuity of copula functions (Nelsen, 2006). Define a measure µ on E2 as µ([−∞, y1 ] × · · · × [−∞, yd ]) := min{ey1 , . . . , eyd }. This measure µ only puts mass on the line {(y1 , . . . , yd ) : ey1 = · · · = eyd }. If ey1 = · · · = eyd =: h, then µ([−∞, y1 ] × · · · × [−∞, yd ]) = h. Clearly, µ is Radon and every set [−∞, y] is a continuity set. Define µt (·) :=  P [((Y1 + t)/a(t) + log k1 , . . . , (Yd + t)/a(t) + log kd ) ∈ ·] , P[Y1 ≤ −t]  then by Lemma 4.12 and (4.23), v  µt (·) → µ(·),  t → ∞.  v  where → is vague convergence. Define a set d  H :=  y ∈ (−∞, ∞] \ U B :  yi ≤ 0 . i=1  109  Then clearly, µ(∂H) = 0 and H is relatively compact. Thus by Theorem 3.2 of Resnick (2007), µt (H) → µ(H),  t → ∞.  That is, letting zi := eyi , as t → ∞, P[W ≤ −dt − a(t) di=1 log ki ] = µt (H) P[Y1 ≤ −t] d  → min ey1 , . . . , eyd :  d  yi ≤ 0 = µLebesgue  log z ≤ 0  z>0:  i=1  = 1.  i=1  which justifies (4.22) and asymptotic additivity holds. Proof of Proposition 4.15: It is well known that under Assumption 4.1 with α > 1, CTEp (Xi ) ∼  α α−1 VaRp (Xi ), p  → 1− , for i = 1, . . . , d (e.g., Zhu and Li, 2012).  By Theorem 2.1 in Asimit et al. (2011), and Proposition 4.13, d  CTEp (Xi ) ∼ i=1  α α−1  d  VaRp (Xi ) ∼ i=1  α VaRp (S) ∼ CTEp (S), α−1  p → 1− ,  which completes the proof. Proof of Proposition 4.16: It is well known that under Assumption 4.2, CTEp (Xi ) ∼ VaRp (Xi ), p → 1− (e.g., Section 3.3.3 of Embrechts et al. (1997)). Also, CTEp (S) ∼ VaRp (S), p → 1− (see (2.15) of Asimit et al. (2011)). Therefore, by Proposition 4.14, d  CTEp (S) ∼ VaRp (S) ∼  d  VaRp (Xi ) ∼ i=1  CTEp (Xi ), i=1  which completes the proof.  110  p → 1− ,  Chapter 5  Tail dependence and conservativity 5.1  Introduction  Dependence modeling with copulas for multivariate random losses plays an important role in accounting for a nonlinear dependence structure in insurance and financial data. For example, the dependence between losses and associated expenses often appears to be asymmetric in upper and lower tails for auto insurance claim data (Frees and Valdez, 1998). There are several important statistical issues for dependence modeling with copulas, such as statistical inference, goodness-of-fit testing and model selection. Among many others, we refer to Joe (1997), Genest and Favre (2007), Genest et al. (2009b) and Brechmann et al. (2012) for statistical issues in copula modeling. When introducing various statistical methodologies into the actuarial community, an important question that should not be avoided is the sensitivity of commonly used risk measures to statistical modeling. As we know, nice properties of statistical methodologies are often derived based on certain nice assumptions. Moreover, there are not sufficient approaches that are tailored for tail risks. It is very challenging for a statistical method to accurately account for tail risks, especially when data in the tail are sparse and rare events leading to large losses happen unexpectedly. Due to the nature of the insurance industry, financial security is par111  ticularly important. Therefore, in practice, actuaries often need to consider more conservative modeling when there is less certainty, and conduct scenario testing accordingly. In Chapter 4, we have studied a concept of tail comonotonicity, which can be a property for absolutely continuous copulas so it is not as restrictive as the comonotonic copula. Upper tail comonotonic copulas have an upper tail dependence parameter of 1 in the sense of the usual tail dependence (Joe, 1997, section 2.1.10), and include the comonotonic copula as well as copulas satisfying the properties of upper comonotonicity (Cheung, 2009) as special cases; we refer to Dhaene et al. (2002a,b) for the former concept, and Dong et al. (2010) and Nam et al. (2011) for further study of the latter concept. For upper comonotonicity, there is a constraint of comonotonicity beyond a certain threshold. But with tail comonotonicity, the constraint applies only for the limit and the data can contribute to the likelihood for the whole support of the copula. For joint non-functional-related losses, tail comonotonicity is a reasonably conservative assumption. The benefit of tail comonotonicity is that it introduces a balance between accuracy and security for assessing dependent tail risks; the less information we have and the larger losses we care about, the more conservative assumptions are adopted. We refer to Chapter 4 for more relevant research on tail comonotonicity. If X1 and X2 are dependent losses, two fundamental conditional specifications E[X1 |X2 > t] and E[X1 |X2 = t] that are relevant to quantitative risk management are the emphasis of this chapter. As t → ∞, for fixed margins, dependence structures will play a role in ordering the above conditional expectations. Sufficient conditions that lead to such a comparison have been derived for bivariate copulas; in particular, for Archimedean copula families, tail behavior of their generators may affect the asymptotic orders, and tail comonotonicity is proved to be conservative in term of the two conditional specifications. For tail comonotonic copulas, second order conditions determine the degree of conservativity, and we propose a way to compare the second order conditions. Other than the two conditional specifications, tail comonotonicity is also conservative in terms of diversification effects, thus we refer to tail comonotonicity as an asymptotically worst dependence structure. Simulations are conducted to further understand the conditions and their influ112  ence on risk measures; tail comonotonicity is shown to be conservative as high-risk scenarios are considered. Finally, an absolutely continuous copula model with tail comonotonicity is applied to an auto insurance claim data; it was used to assess the magnitude of associated expenses based on the information of loss. This chapter is organized as following: Section 5.2 reports results on the two conditional specifications and second order properties of tail comonotonicity. The asymptotically worst dependence structure is briefly introduced in Section 5.3. Some findings of the simulation study are presented in Section 5.4, and a data analysis with tail comonotonicity is conducted in Section 5.5. Section 5.6 concludes with discussions, and the proofs are collected in Section 5.7.  5.2  Conditional specifications  For risk management in actuarial science or quantitative finance, one often concerns the influence of risks from an individual asset over the whole portfolio. If (X1 , . . . , Xd ) are unbounded dependent losses, then conditional tail expectation of the forms E[  d i=1 Xi |Xk  > t] or E[  d i=1 Xi |Xk  = t] can be used as relevant  risk measures, where t is usually a high quantile of Xk . Due to linearity of probability expectations, it suffices to study the conditional tail expectation of the forms E[Xi |Xk > t] or E[Xi |Xk = t] for a given pair of i, k ∈ {1, . . . , d}. For notational convenience, they are written as the following two forms: E[X1 |X2 > t];  (5.1)  E[X1 |X2 = t].  (5.2)  Moreover, these forms of conditional specifications may also be used to model a univariate time series, such as, E[Xi+1 |Xi > t] where copulas can be used to model the serial dependence between Xi and Xi+1 (Patton, 2009). The study of these two forms in the literature has been mainly on the influence of dependence and asymptotic analysis for the threshold t in (5.1) or (5.2) goes to infinity. For example, Landsman and Valdez (2003) has a formula for (5.1) when (X1 , X2 ) follows a bivariate elliptical distribution; Zhu and Li (2012), and Chapter 6 conduct asymptotic analysis under the condition of (second order) regular  113  variation. In this section, we will investigate the effects of dependence structures between X1 and X2 on these two conditional specifications: E[X1 |X2 > t] and E[X1 |X2 = t] as t → +∞. In particular, we will study how conservative tail comonotonicity is for modeling dependent risks. From Chapters 3 and 4, we know conditions for Archimedean copulas to be tail comonotonic or satisfy other upper/lower tail properties. Therefore, we illustrate general results in the special case of Archimedean copulas. In order to illustrate the conservativity of tail comonotonicity, we will focus on the bivariate case although tail comonotonicity is a multivariate concept. For multivariate extensions, the conditional specification can be of the following forms: E[X1 |X2 > t1 , . . . , Xd > td ] or E[X1 |X2 = t1 , . . . , Xd = td ]. These two forms may be useful in quantitative risk management for portfolios or constructing models for multiple regression analysis. However, these multivariate extensions are harder to analyze and are left for future study. In this section, X1 , X2 , Y1 and Y2 are assumed to be absolutely continuous random variables defined on [0, ∞), Xi , Yi ∼ Fi , i = 1, 2, written as X, Y ∈ R(F1 , F2 ), the Fr´echet space of all bivariate random vectors whose marginal distributions are F1 and F2 , respectively, and F1 , F2 are absolutely continuous and defined on [0, ∞). Furthermore, E[Xi ] = E[Yi ] < ∞, i = 1, 2. Then sufficient conditions that lead to the following two asymptotic inequalities will be studied. Relation (I):  E[X1 |X2 > t]  E[Y1 |Y2 > t],  t → ∞;  (5.3)  Relation (II):  E[X1 |X2 = t]  E[Y1 |Y2 = t],  t → ∞.  (5.4)  In our analysis of the above two conditions, the limits of the conditional distributions of bivariate copulas will appear, specifically C1|2 (u|1) and C1|2 (u|0) for 0 < u < 1. If (U, V ) ∼ C, these are the limiting distributions of [U |V = v] as v → 1− or v → 0+ . These limiting conditional distributions appear in the tail dependence results in Cooke et al. (2011). With this form of dependence in the tails, C1|2 (u|1) = 0 for 0 ≤ u < 1 or [U |V = 1] degenerates at 1 is the strongest upper tail dependence and C1|2 (u|0) = 1 for 0 < u ≤ 1 or [U |V = 0] degenerates at 0 is the strongest lower tail dependence. Note that, this tail dependence form is different from the widely-used form 114  of tail dependence defined in Joe (1997, section 2.1.10); we refer to the latter as “usual tail dependence” in this chapter. Next consider the special case of a bivariate Archimedean copula constructed by a LT ψ. The condition C1|2 (u|1) = 0 for 0 ≤ u < 1 occurs when ψ (0) = −∞; this is also the condition to achieve the usual upper tail dependence. The condition C1|2 (u|0) = 1 for 0 < u ≤ 1 occurs as 0 < r < 1 under the mild assumption ψ(s) ∼ T (s) = a1 sq exp{−a2 sr } with a1 > 0, a2 ≥ 0, and ψ (s) = T (s), as s → ∞; this matches conditions to achieve either lower tail dependence or intermediate lower tail dependence, i.e., a lower tail order of 1 ≤ κ < 2.  5.2.1  The case: E[X1 |X2 > t]  Proposition 5.1 Suppose X := (X1 , X2 ), Y := (Y1 , Y2 ) and X, Y ∈ R(F1 , F2 ). ∗ (u|v)/C (u|v)] < 1 (a) If X ∼ C ∗ (F1 , F2 ), Y ∼ C(F1 , F2 ) and limv→1− [C1|2 1|2  for all 0 < u < 1, then Relation (I) holds. (b) If C ∗ and C are survival copulas of ∗ (u|v)]/[1 − C ∗ and C, X ∼ C ∗ (F1 , F2 ), Y ∼ C(F1 , F2 ), and limv→0+ [1 − C1|2  C1|2 (u|v)] < 1 for all 0 < u < 1, then Relation (I) holds. For constructing Relations (I) and (II) for Archimedean copulas, the upper and lower tail behavior of the following function is critical: Υ(u) :=  ϕ (ϕ−1 (u)) , ψ (ψ −1 (u))  u ∈ [0, 1],  (5.5)  where ψ and ϕ are LTs for the copulas or survival copulas of X and Y. Note that Υ(0) and Υ(1) are defined as the corresponding limits, provided that the limits exist. Corollary 5.2 Suppose X := (X1 , X2 ), Y := (Y1 , Y2 ), X, Y ∈ R(F1 , F2 ), and their dependence structures are C ∗ and C that are bivariate Archimedean copulas constructed as in (2.4) with LTs ψ and ϕ, respectively. If Υ(1) exists and Υ(u) > Υ(1) for any 0 < u < 1, then Relation (I) holds. Remark 5.1 The condition (5.12) in the proof of Proposition 5.1 holds if C ∗ is larger than C in the PQD or concordance ordering (Joe, 1997; Shaked and Shanthikumar, 2007). The concordance ordering holds for copulas from a single com115  monly used 1-parameter bivariate copula family, but Proposition 5.1 and Corollary 5.2 can also be applied to two copulas from two different parametric families, as shown in Example 5.1. Remark 5.2 The condition Υ(u) > Υ(1) is the same as ξ (s) > ξ (0), where ξ = ψ −1 ◦ ϕ. For concordance ordering of the two Archimedean copulas, a sufficient condition is ξ convex or ξ (s) increasing in s (Joe, 1997, Corollary 4.2). So the tail comparison here is satisfied with the weaker condition. Example 5.1 (a) (bivariate MTCJ vs Gumbel). The MTCJ copula is based on the gamma LT (1 + s)−1/δM for δM > 0 and the Gumbel copula is based on the positive stable LT exp{−s−1/δG } for δG > 1. Suppose the parameters for MTCJ and Gumbel copulas are δM and δG , respectively, and the LTs are ϕ and ψ, respectively. Then Υ(u) = (δG /δM )uδM (− log u)δG −1 , and Υ(u) > Υ(1) = 0 for any 0 < u < 1. So if (X1 , X2 ) has an arbitrary Gumbel copula and (Y1 , Y2 ) has an arbitrary MTCJ copula, and Xi , Yi ∼ Fi , i = 1, 2, Relation (I) must hold. This case is not surprising as the Gumbel copula has upper tail dependence for any δG > 1 and the MTCJ has upper tail order of 2 (upper tail independence) for any δM > 0. That is, the Gumbel copula has more probability in the upper tail and ∗  C (u, v; δG ) ≥ C(u, v; δM ) or equivalently C ∗ (u, v; δG ) ≥ C(u, v; δM ) for (u, v) in an upper set SδG ,δM that can include most of [0, 1] × [v0 , 1] for some v0 near 1. (b) (Joe/B5 vs Gumbel). The Joe/B5 copula is based on the Sibuya LT 1 − (1 − e−s )1/δJ  for δJ > 1. This also has upper tail dependence so the comparison with  Gumbel requires a careful check of the conditions in Proposition 5.1. Let C Joe and C Gum denote these two copula families. Using the conditional distributions on page 147 of Joe (1997), as v → 1− , Joe (u|v; δ ) C1|2 J Gum (u|v; δ ) C1|2 G  ∼  (1 − u)−δJ +1 {1 − (1 − u)δJ } · (1 − v)δG −δJ . u(− log u)−δG +1  The limit is 0 if δG > δJ and ∞ if δG < δJ and finite if δG = δJ . In the latter case, the ratio is shown numerically to be > 1 for 0 < u < 1. Therefore if δG = δJ = δ > 1 and C ∗ = C Gum (; δ), C = C Joe (; δ), (X1 , X2 ) ∼ C ∗ (F1 , F2 ), (Y1 , Y2 ) ∼ C(F1 , F2 ), then Relation (I) must hold. 116  For applications in insurance and finance, we often need to consider a copula that has upper tail dependence. In Table 5.1, we summarize the comparison in the sense of Corollary 5.2 for some commonly used upper tail dependent copulas. Table 5.1: Comparisons between Gumbel, Joe, BB1 and BB7 copula families Copula Gumbel Joe BB1 1 BB7  Parameter(s) 1≤δ 1≤δ 1 ≤ δ; 0 < θ 1 ≤ δ; 0 < θ  ψδ (ψδ−1 (u)) − 1δ u(− log u)1−δ − 1δ (1 − u)1−δ [1 − (1 − u)δ ] 1 θ+1 − θδ u (u−θ − 1)1−δ 1 − θδ (1 − u)1−δ [1 − (1 − u)δ ]1+θ  Domination term as u → 1− (− log u)1−δ ∼ (1 − u)1−δ (1 − u)1−δ (u−θ − 1)1−δ ∼ θ1−δ (1 − u)1−δ (1 − u)1−δ  1. The notation of δ, θ is exchanged from their original form in Joe (1997).  Note that the upper tail dependence parameters for all of the copulas in Table 5.1 are 2 − 21/δ . The following result tells us how to use Table 5.1. It includes part (b) of Example 5.1 as a special case. Corollary 5.3 Suppose X := (X1 , X2 ), Y := (Y1 , Y2 ), X, Y ∈ R(F1 , F2 ), and their dependence structures are copulas C ∗ , C ∈ {Gumbel, Joe, BB1, BB7}. If δ ∗ > δ, that is, the upper tail dependence parameters λ∗ > λ, then Relation (I) holds. For modeling dependent risks, we often need to reflect a copula to get its survival copula. Now we consider the condition in the sense of Corollary 5.2 when survival Archimedean copulas are used to model the risks. Proposition 5.4 Suppose X := (X1 , X2 ), Y := (Y1 , Y2 ), X, Y ∈ R(F1 , F2 ), X ∼ C ∗ (F1 , F2 ) and Y ∼ C(F1 , F2 ), where C ∗ and C are Archimedean copulas constructed as in (2.4) with LTs ψ and ϕ, respectively. If ψ −1 ◦ ϕ(s) is strictly convex for large s, and in particular, if Υ(u) is strictly decreasing in u near 0, then Relation (I) holds. Remark 5.3 The above result is on the ordering in the lower tail of two copulas. For the concordance ordering of Archimedean copulas Cψ and Cϕ , a sufficient condition is that ψ −1 ◦ ϕ is superadditive (Joe, 1997, Theorem 4.1), which holds if ψ −1 ◦ ϕ is convex due to ψ −1 (ϕ(0)) = 0 (Marshall and Olkin, 2007, Proposition 117  A.11). Obviously, when ψ −1 ◦ ϕ is superadditive, Relation (I) must hold. But the condition in Proposition 5.4 only requires the superadditive inequality where one argument is large and the other is arbitrary, and this can be checked via the behavior of Υ(u) as u → 0+ . From Corollary 5.2 and Proposition 5.4, the tail behavior of ψ (ψ −1 (u)) for u → 1− and u → 0+ determine the upper and lower tail association in the sense of Relation (I). So we list in Table 5.2 the conditions for commonly used 1- or 2-parameter Archimedean copulas that appear in Joe (1997). By looking at the conditions listed in Table 5.2, we can easily compare the conditional tail expectations in the sense of Proposition 5.4 (or Corollary 5.2). Those conditions will also be used in the next subsection for establishing Relation (II). Example 5.2 (BB3 vs BB2) As u → 0+ , observed from Table 5.2, the term −θ −1)  e−δ(u  θ  for BB2 and the term e−δ(− log u) for BB3 dominate the condition in  Proposition 5.4 for comparing these two copulas. It is clear that the former term goes to 0 much faster than the latter as u → 0+ . Therefore, if C ∗ is BB2 and C is BB3, and C ∗ and C are copulas for X and Y respectively, then Relation (I) holds. It is also consistent to the fact that BB2 is more lower tail positive dependent than BB3 below a sufficiently low threshold, although they are both lower tail comonotonic (see Chapter 4). Table 5.2 shows that ψ (ψ −1 (0)) ≡ 0 for all of the Archimedean copula families listed, which reminds us of the following fact: suppose ψ(s) is the LT of a positive random variable, then ψ (n) (s) → 0 as s → ∞ for n = 1, 2, . . . . The following lemmas are the Monotone Density Theorem (Bingham et al., 1987; Embrechts et al., 1997) applied to slowly varying functions. Lemma 5.5 Let ∈ RV0 and (s) =  s 0  (u)du (or (s) =  is ultimately monotone as s → ∞, then (s) = Lemma 5.6 Let ∈ RV0 (0+ ) and (s) =  s 0  o(s−1  (u)du). If (s)  (s)), s → ∞.  (u)du. If (s) is ultimately mono-  tone as s → 0+ , then (s) = o(s−1 (s)), s → 0+ .  118  ∞ s  Table 5.2: Condition ψ (ψ −1 (u)) for parametric Archimedean copula families Copula Frank(B3) MTCJ(B4) Joe(B5) Gumbel(B6) BB1 BB2 BB3 BB6 1 BB7 BB8 BB9 BB10  Parameter(s) 0≤δ 0≤δ 1≤δ 1≤δ 1 ≤ δ; 0 < θ 0 < δ, θ 0 < δ; 1 ≤ θ 1 ≤ δ, θ 1 ≤ δ; 0 < θ 0 < δ ≤ 1; 1 ≤ θ 0 ≤ α; 1 ≤ θ 0 < α; 0 ≤ θ ≤ 1  ψ (ψ −1 (u)) (1/δ)(1 − eδu ) −(1/δ)u1+δ −(1/δ)(1 − u)1−δ [1 − (1 − u)δ ] −(1/δ)u(− log u)1−δ −(1/(θδ))uθ+1 (u−θ − 1)1−δ −θ −(1/(θδ))u1+θ e−δ(u −1) θ −(1/(θδ))u(− log u)1−θ e−δ(− log u) 1−θ θ −(1/(θδ))(1 − u) [1 − (1 − u) ][− log[1 − (1 − u)θ ]]1−δ 1−δ −(1/(θδ))(1 − u) [1 − (1 − u)δ ]1+θ 1−θ −(1/(θδ))(1 − δu) + (1/(θδ))(1 − δu) −(1/θ)u(α − log u)1−θ −αu − αθ(1 − θ)−1 u1+1/α  1. The notation of δ, θ is exchanged from their original form in Joe (1997).  The following two results establish that tail comonotonicity is conservative in the sense of Relation (I). By Proposition 4.3, the condition 1 − ψ ∈ RV0 (0+ ) in Corollary 5.7 implies that the Archimedean copula Cψ is upper tail comonotonic, and the condition ψ ∈ RV0 in Corollary 5.8 implies that the Archimedean copula Cψ is lower tail comonotonic. Corollary 5.7 Suppose X := (X1 , X2 ), Y := (Y1 , Y2 ), X, Y ∈ R(F1 , F2 ), and their dependence structures are C ∗ and C that are Archimedean copulas constructed as in (2.4) with LTs ψ and ϕ, respectively. If 1 − ψ ∈ RV0 (0+ ) and lim inf s→0+ (−sϕ (s))/(1 − ϕ(s)) > 0, then Relation (I) holds. Corollary 5.8 Suppose X := (X1 , X2 ), Y := (Y1 , Y2 ), X, Y ∈ R(F1 , F2 ), X ∼ C ∗ (F1 , F2 ) and Y ∼ C(F1 , F2 ), where C ∗ and C are Archimedean copulas constructed as in (2.4) with LTs ψ and ϕ, respectively. If Υ(u) is ultimately monotone as u → 0+ , ψ ∈ RV0 and lim inf s→∞ (−sϕ (s))/(ϕ(s)) > 0, then Relation (I) holds.  119  Remark 5.4 The conditions lim inf (−sϕ (s))/(1 − ϕ(s)) > 0 (for copula upper tail), s→0+  lim inf (−sϕ (s))/ϕ(s) > 0 s→∞  (for copula lower tail)  cover tail dependence, intermediate tail dependence and tail orthant independence cases. For instance, for the case of lower tail, let π := − lims→∞ (sϕ (s))/ϕ(s), then π = 0 is tail comonotonicity, 0 < π < ∞ corresponds to tail dependence and π = ∞ leads to intermediate tail dependence or near independence (tail order κ = d). We refer to Charpentier and Segers (2009) and Chapter 3 for a comprehensive study of tail behavior of Archimedean copulas.  5.2.2  The case: E[X1 |X2 = t]  In this subsection, we have some parallel results to the previous subsection by conditioning on X2 = t instead of X2 > t. Proposition 5.9 Suppose X := (X1 , X2 ), Y := (Y1 , Y2 ) and X, Y ∈ R(F1 , F2 ). ∗ (u|v)/C (u|v)] < 1 (a) If X ∼ C ∗ (F1 , F2 ), Y ∼ C(F1 , F2 ) and limv→1− [C1|2 1|2  for all 0 < u < 1, then Relation (II) holds. (b) If C ∗ and C are survival copulas of ∗ (u|v)]/[1 − C ∗ and C, X ∼ C ∗ (F1 , F2 ), Y ∼ C(F1 , F2 ) and limv→0+ [1 − C1|2  C1|2 (u|v)] < 1 for all 0 < u < 1, then Relation (II) holds. Corollary 5.10 Suppose X := (X1 , X2 ), Y := (Y1 , Y2 ), X, Y ∈ R(F1 , F2 ), and their dependence structures are C ∗ and C that are Archimedean copulas constructed as in (2.4) with LTs ψ and ϕ, respectively. If Υ(1) exists and Υ(u) > Υ(1) for any 0 < u < 1, then Relation (II) holds. Corollary 5.11 Suppose X := (X1 , X2 ), Y := (Y1 , Y2 ), X, Y ∈ R(F1 , F2 ), X ∼ C ∗ (F1 , F2 ) and Y ∼ C(F1 , F2 ), where C ∗ and C are Archimedean copulas constructed as in (2.4) with LTs ψ and ϕ, respectively. If ψ −1 ◦ ϕ(s) is strictly convex for large s, and in particular if for u sufficiently small, Υ(u) is strictly decreasing in u, then Relation (II) holds.  120  Corollary 5.12 Suppose X := (X1 , X2 ), Y := (Y1 , Y2 ), X, Y ∈ R(F1 , F2 ), and their dependence structures are C ∗ and C that are Archimedean copulas constructed as in (2.4) with LTs ψ and ϕ, respectively. If 1 − ψ ∈ RV0 (0+ ) and lim inf s→0+ (−sϕ (s))/(1 − ϕ(s)) > 0, then Relation (II) holds. Corollary 5.13 Suppose X := (X1 , X2 ), Y := (Y1 , Y2 ), X, Y ∈ R(F1 , F2 ), X ∼ C ∗ (F1 , F2 ) and Y ∼ C(F1 , F2 ), where C ∗ and C are Archimedean copulas constructed as in (2.4) with LTs ψ and ϕ, respectively. If Υ(u) is ultimately monotone, ψ ∈ RV0 and lim inf s→∞ (−sϕ (s))/(ϕ(s)) > 0, then Relation (II) holds.  5.2.3  Second order conditions and conservativity  From Example 5.2, we notice that BB2 in general has a stronger positive association in the lower tail than BB3, although BB2 and BB3 are both lower tail comonotonic. In this subsection, we will study second order conditions under which tail comonotonicity may appear various strength of lower tail dependence. Suppose that copula C is lower tail comonotonic, if there exist auxiliary functions A(t), B(t) such that B lim  u→0+  C(uw1 ,...,uwd ) u  − min(w1 , . . . , wd ) A(u)  = H(w1 , . . . , wd ) ≡ 0,  w > 0, (5.6)  then the speed of convergence can be characterized by the auxiliary functions A(t) and B(t). Second order conditions may play an important role in modeling high-risk scenarios. We refer to Degen et al. (2010) and Chapter 6 for relevant studies under the condition of second order regular variation (de Haan and Stadtm¨uller, 1996). The idea here is similar to second order regular variation, but the auxiliary functions A(t) and B(t) may have various forms other than a power function as in the second order regular variation. Lemma 5.14 Let C be a d-dimensional Archimedean copula constructed with LT ψ ∈ RV0 as in (2.4). Letting w∗ := min{w1 , . . . , wd }, and w∗ is unique in w, 121  then as u → 0+ , C(uw1 , . . . , uwd ) ∼ uw∗ + ψ (ψ −1 (uw∗ )) ψ −1 (uwi ) =: uw∗ + (u, w),  × i∈{j:wj =w∗ }  where the second order term (u, w) = o(u), u → 0+ .  Based on Lemma 5.14, one can obtain concrete forms of second order terms. The following are examples for the bivariate case of BB2 and BB3, for which the auxiliary functions have simple forms. The derivation is straightforward by using Lemma 5.14, thus omitted. Example 5.3 (BB2) The auxiliary functions for BB2 copula are A(t) = t−θ , θ > 0, B(t) = log |t|; that is, log lim  u→0+  Cψ (uw1 ,uw2 ) − u u−θ  min(w1 , w2 )  =: H(w1 , w2 ) = −δ|w1−θ − w2−θ |.  Therefore, θ is the second order parameter that dominates the speed of convergence. Example 5.4 (BB3) The auxiliary functions for BB3 copula are A(t) = (− log t)θ−1 , θ > 1, B(t) = log |t|; that is, log lim  u→0+  Cψ (uw1 ,uw2 ) u  − min(w1 , w2 )  (− log u)θ−1  =: H(w1 , w2 ) = −δθ| log w1 − log w2 |.  122  So θ is the second order parameter. For statistical inference with BB2 or BB3 in the lower tail, if we want to reduce the number of the parameters we can also consider to simply assume δ to be a fixed number, say δ = 0.01 or δ = 0.1. From computational experiments, we find that there are some redundancies when we involve the parameter δ. For both BB2 and BB3, when θ is larger, both u−θ and (− log u)θ−1 converges to +∞ faster as u → 0+ and the copula has more positive dependence at the lower sub-extreme. By looking at the auxiliary functions of BB2 and BB3, we may also conclude that BB2 is more lower tail positive dependent than BB3 below a sufficiently low threshold, which is consistent to Example 5.2. The reason is due to u−θBB2 goes to +∞ faster than (− log u)θBB3 −1 as u → 0+ , and this just implies that the convergence of CBB2 (uw1 , uw2 )/u → min{w1 , w2 } is faster than the convergence of CBB3 (uw1 , uw2 )/u → min{w1 , w2 }, as u → 0+ .  5.3  Asymptotically worst dependence structures  As we have shown in Section 5.2, tail comonotonicity affects the conditional specifications. Moreover, tail comonotonicity may also influence risk measures for d  aggregate losses. For instance, if Xi = Yi , i = 1, . . . , d, and the dependence structures of (X1 , . . . , Xd ) and (Y1 , . . . , Yd ) are different, then risk measures on and  Xi  Yi may be different, although they have the same marginal distributions.  In this section, we will briefly study an asymptotically worst dependence concept brought about by tail comonotonicity. Let X = (X1 , . . . , Xd ) be a d-dimensional real-valued random vector, with univariate cdf’s F1 , . . . , Fd . The Fr´echet space containing all the random vectors that possess these margins is denoted as R(F1 , . . . , Fd ). Suppose g : Rd → R is an aggregate function, and Qp : R → R+ is a quantile based risk measure, where p is the probability level and is usually close to 1. Then X is said to have an asymptotically worst dependence structure in terms of the risk measure Qp and the aggregate function g, if for any X ∈ R(F1 , . . . , Fd ), Qp [g(X)]  Qp [g(X )],  123  p → 1− .  (5.7)  Note that, in Mainik and R¨uschendorf (2012), an asymptotic ordering called asymptotic portfolio loss order has been used to compare the tail probabilities of two portfolios. In this section, we will study situations where tail comonotonicity becomes an asymptotically worst dependence structure. For the choice of risk measures, we restrict ourself to VaR and CTE. For aggregate functions, we only consider the case where g(x1 , . . . , xd ) = d i=1 xi .  Then this concept is also relevant to conservative assessments for diver-  sification effects of risk aggregation.  5.3.1  Conditional tail expectation  Assumption 5.1 Let X be an upper tail comonotonic non-negative random vector with continuous marginal distributions F1 , . . . , Fd defined on [0, ∞) such that 1 − Fi (t) ∈ RV−α with α > 1, and lim [1 − Fi (t)]/[1 − F1 (t)] = ki ,  t→∞  0 < ki < ∞  for any i = 1, . . . , d. The next result shows that tail comonotonicity is an asymptotically worst dependence structure for tail equivalent random variables that have regularly varying upper tails. Proposition 5.15 If X satisfies Assumption 5.1, then for any X ∈ R(F1 , . . . , Fd ), CTEp (X1 + · · · + Xd )  CTEp (X1 + · · · + Xd ),  p → 1− .  In this sense, upper tail comonotonicity provides a conservative dependence structure.  Assumption 5.2 Let X be an upper tail comonotonic non-negative random vector with continuous marginal distributions F1 , . . . , Fd defined on [0, ∞) and there  124  exists a positive measurable function a(·) such that 1 − Fi (t + a(t)s) = es , t→∞ 1 − Fi (t) lim  ∀s ∈ R,  (5.8)  and limt→∞ [1 − Fi (t)]/[1 − F1 (t)] = ki with 0 < ki < ∞ for any i = 1, . . . , d. Due to Proposition 4.16, we also have the following result. Proposition 5.16 If X satisfies Assumption 5.2, then for any X ∈ R(F1 , . . . , Fd ), CTEp (X1 + · · · + Xd )  5.3.2  CTEp (X1 + · · · + Xd ),  p → 1− .  Value at risk  Based on Propositions 4.13 and 4.14, it suffices to find conditions on the dependence structure for X, where the margins are specified by either Assumptions A or B, so that VaRp becomes subadditive as p is in a left-neighbor of 1. It is well known that, in general VaR is not subadditive. However, with certain regularity conditions, VaR may also be subadditive; the study of the regularity conditions has been an important topic in the literature but is out of the scope of this chapter. In what follows, some results for asymptotically worst dependence structure in terms of VaR will be given. Those are based on known facts of situations where VaR becomes (asymptotically) subadditive. Proposition 5.17 Let X = (X1 , . . . , Xd ) ∈ MRVd (−β), β > 1, with identically d  distributed margins on [0, ∞), and X is upper tail comonotonic with Xi = Xi , i = 1, . . . , d. Then VaRp (X1 + · · · + Xd )  VaRp (X1 + · · · + Xd ),  p → 1− .  For marginal distributions that are in the Maximum Domain of Attraction of Gumbel, subadditivity of VaR may hold for multivariate elliptical distributions (McNeil et al., 2005, Theorem 6.8). So the following result holds.  125  Proposition 5.18 Let X = (X1 , . . . , Xd ) be a d-dimensional elliptical distribution where the margins satisfying (5.8) with support (−∞, ∞), and X is upper tail d  comonotonic with Xi = Xi , i = 1, . . . , d. Then VaRp (X1 + · · · + Xd )  5.4  VaRp (X1 + · · · + Xd ),  p → 1− .  Simulation study  In this section, we report on simulations to illustrate the conservativity of tail comonotonicity in terms of E[X1 |X2 > VaRp (X2 )], E[X1 |X2 = VaRp (X2 )], VaRp (X1 + X2 ) and CTEp (X1 + X2 ) as p → 1− , respectively.  5.4.1  Conditional specifications  Now we compare the effects on E[X1 |X2 > VaRp (X2 )] and E[X1 |X2 = VaRp (X2 )] for different Archimedean copulas. Comonotonicity and the survival copulas of BB2, MTCJ, Gumbel and Frank (denoted as s.BB2, s.MTCJ, s.Gumbel and s.Frank) are used to model the dependence structures of two margins X1 and X2 ; that is, the strength of upper tail positive dependence are different. From Chapter 3, considering both tail order and tail dependence parameters, the strength of dependence in the upper tail can be ordered as: Comonotonicity  s.BB2  s.MTCJ  s.Gumbel  s.Frank.  (5.9)  Choosing parameters for these Archimedean copulas to have the same Blomqvist’s β (since MLEs for different copula families often lead to a similar Blomqvist’s β), we can compare the values of E[X1 |X2 > VaRp (X2 )] or E[X1 |X2 = VaRp (X2 )] for different Archimedean copulas. Figures 5.1 and 5.2 contain the plots for the two conditional specifications, respectively, with p ranging from 80% to 99%; this range is most relevant for applications. The calculations were based on numerical integration through Monte Carlo simulations. For both figures, the copula parameters used in the simulation were the following: BB2: δ = 2, θ = 0.4; MTCJ: δ = 1.424; Gumbel: δ = 1.729 and Frank: δ = 3.844. The common Blomqvist’s β was 0.421. 126  From Figure 5.1, it is interesting that, when p is sufficiently large (say, p ≥ 80% for the plots), the order of the values of E[X1 |X2 > VaRp (X2 )] is the same as the order in (5.9). Moreover, the same pattern can be observed regardless of the margins. The pattern for Figure 5.2 is different than that for E[X1 |X2 > VaRp (X2 )]. But when p is sufficiently large (say, p ≥ 95% in this example), the rank of E[X1 |X2 = VaRp (X2 )] is still kept as the same as (5.9). Based on these comparisons, the BB2 tail comonotonic copula is reasonable and better than the usual comonotonicity when considering a relatively lower probability level p where more data are available. However, as less data are available when considering a higher probability level, the tail comonotonic copula becomes more conservative and approaches the upper bound provided by the usual comonotonicity. This kind of conservatism makes more sense for quantitative risk management.  5.4.2  Asymptotically worst dependence structures  To illustrate the idea of applying tail comonotonicity for a reasonably conservative modeling of high-risk scenarios, we generated a bivariate data with s.BB1 as the copula and standard Pareto as the margins. We transformed the data to rank scores and fitted these bivariate rank scores by copulas that possess upper tail dependence, such as t, Gumbel, s.BB2 and s.BB3 copulas. The estimation was based on maximum quasi-likelihood estimation and the margins are assumed to be given. The Monte Carlo method was then used to approximate the corresponding VaRs and CTEs, based on the fitted copulas and the known margins. Table 5.3 and 5.4 present comparisons between different copulas, for the cases of more dependence and less dependence, respectively. The margins were standard Pareto distributions with cdf F (x) = 1 − x−3 . For each run, the sample size was 105 and “se”, the simulation error for the Monte Carlo method was based on 1, 000 runs. In the tables, “ind” is for the independent case, “s.BB1.o” means s.BB1 is the true model, and λ is the corresponding upper tail dependence parameter. Table 5.3 shows that: (1) Overlooking the dependence between margins will seriously underestimate the aggregate risks. (2) The Gumbel copula has the smallest AIC, but it underestimates the risks. (3) The BB2 and BB3 copulas overesti-  127  Figure 5.1: The value of y-axis is E[X1 |X2 > VaRp (X2 )]. In the first plot, X1 , X2 have Exponential distributions with cdf F (x) = 1 − Exp(−x/σ). In the second plot, X1 , X2 have Pareto distributions with cdf F (x) = 1 − (1 + x/σ)−θ . Exponential margins with σ = 10  Comonotonicity s.BB2 s.MTCJ s.Gumbel s.Frank  E( X1 | X2 > t )  50  40  30  20  0.80  0.85  0.90  0.95  p  Pareto margins with θ = 3, σ = 20  120  Comonotonicity s.BB2 s.MTCJ s.Gumbel s.Frank  E( X1 | X2 > t )  100  80  60  40  20 0.80  0.85  0.90  p  128  0.95  Figure 5.2: The value of y-axis is E[X1 |X2 = VaRp (X2 )]. In the first plot, X1 , X2 have Exponential distributions with cdf F (x) = 1 − Exp(−x/σ). In the second plot, X1 , X2 have Pareto distributions with cdf F (x) = 1 − (1 + x/σ)−θ . Exponential margins with σ = 10  Comonotonicity s.BB2 s.MTCJ s.Gumbel s.Frank  E( X1 | X2 = t )  40  30  20  0.80  0.85  0.90  0.95  p  Pareto margins with θ = 3, σ = 20  Comonotonicity s.BB2 s.MTCJ s.Gumbel s.Frank  E( X1 | X2 = t )  60  40  20  0.80  0.85  0.90  p  129  0.95  mate the risk a little bit and could provide a reasonable conservative estimate for this case, especially when the probability level p is very high. (4) The tail dependence parameters seem relevant to the magnitude of the aggregate risks: a higher tail dependence parameter tends to lead to a larger aggregate risk, especially when the probability level is quite high. This also suggests that tail comonotonicity is conservative under some conditions. Comparing Tables 5.3 and 5.4, we can also find that overestimation arising from tail comonotonicity is milder for the case of stronger tail dependence than for the case of less tail dependence. Table 5.3: (More dependence) VaR and CTE for X1 + X2 . The MLEs were based on the whole sample generated from s.BB1 (δ = 1.57, θ = 1.68, λ = 0.77, more dependence) with a sample size of 2000. The bold AIC value is the smallest.  ind se s.BB1.o se t se Gumbel se s.BB2 se s.BB3 se  VaR90 4.084 <0.001 4.270 <0.001 4.269 <0.001 4.272 <0.001 4.279 <0.001 4.263 <0.001  VaR995 8.949 0.003 11.573 0.005 11.275 0.005 11.537 0.005 11.616 0.005 11.643 0.005  CTE90 5.532 0.001 6.400 0.001 6.315 0.001 6.389 0.001 6.419 0.001 6.416 0.001  CTE995 12.631 0.010 17.369 0.017 16.842 0.016 17.289 0.016 17.411 0.017 17.440 0.016  AIC -  λ -  -  0.77  -2678.4  0.53  -2887.7  0.73  -2737.6  1  -2782.7  1  With the same simulation settings, we repeated the experiment for 40 times. Table 5.5 is a summary of these simulations. The approach is conservative only when the probability level is very high (say, > 99%). With a moderately high probability level, say, 90%, this approach would be not conservative. Moreover, tail comonotonicity worked better for the case where the original dependence is actually stronger.  130  Table 5.4: (Less dependence) VaR and CTE for X1 + X2 . The MLEs were based on the whole sample generated from s.BB1 (δ = 2, θ = 0.4, λ = 0.42, less dependence) with a sample size of 2000. The bold AIC value is the smallest.  ind se s.BB1.o se t se Gumbel se s.BB2 se s.BB3 se  5.5  VaR90 4.083 <0.001 4.252 <0.001 4.262 <0.001 4.245 <0.001 4.234 <0.001 4.217 <0.001  VaR995 8.951 0.004 11.094 0.005 11.100 0.005 11.417 0.006 11.480 0.005 11.192 0.005  CTE90 5.531 0.001 6.249 0.001 6.261 0.001 6.331 0.001 6.345 0.001 6.220 0.001  CTE995 12.641 0.010 16.515 0.015 16.465 0.015 17.105 0.015 17.225 0.016 16.994 0.017  AIC -  λ -  -  0.42  -2099.0  0.39  -1946.5  0.65  -1574.3  1  -2054.5  1  Application on a claim dataset  We now illustrate how tail comonotonicity can be used in real applications with the dataset of Loss (X2 ) versus Allocated Loss Adjustment Expense (ALAE) (X1 ) that has been studied in the literature such as Frees and Valdez (1998), and Klugman and Parsa (1999). The margins will be fitted by a Pareto distribution with the following distribution function (same as in Frees and Valdez (1998)) F (x) = 1 − (1 + x/σ)−θ . Its density and inverse functions are f (x) = (θ/σ) (1 + x/σ)−θ−1 and F −1 (x) = σ(1 − x)−1/θ − σ. In addition to the Gumbel copula fitted in Frees and Valdez (1998), we use s.BB2 as a conservative model to fit the dependence structure as well. ALAE for auto insurance claims often involves legal expenses that could be very large. Just like the current dataset, for a relatively small amount of loss, the ALAE can be very large and there are usually no limits that can be set for the expense. If one wants to assess the risk of future total loss and expense, one way is to think that the historical dataset reflects the future claims. However, this method may be too optimistic and does not account for the uncertainty beyond the historical dataset. 131  Table 5.5: VaR and CTE for X1 + X2 . The first 4 columns of values are the means of corresponding quantities calculated from 40 random samples generated from the same settings as before. The rest are frequency of those quantities being greater than those for s.BB1.o. (A: more dependent, large sample size; B: less dependent, large sample size; C: more dependent, small sample size; D: less dependent, small sample size. More dependence: δ = 1.57, θ = 1.68, λ = 0.77; less dependence: δ = 2, θ = 0.4, λ = 0.42. Large sample: N = 2000; small sample: N = 50.)  A s.BB1.o t Gumbel s.BB2 s.BB3 B s.BB1.o t Gumbel s.BB2 s.BB3 C s.BB1.o t Gumbel s.BB2 s.BB3 D s.BB1.o t Gumbel s.BB2 s.BB3  VaR90 VaR995 CTE90 CTE995 VaR90 VaR995 CTE90 CTE995 4.270 11.574 6.400 17.365 4.266 11.281 6.313 16.845 2 0 0 0 4.270 11.536 6.387 17.303 15 0 0 0 4.277 11.612 6.417 17.423 40 40 40 40 4.249 11.643 6.408 17.490 0 40 33 40 4.251 11.090 6.247 16.510 4.257 11.093 6.255 16.488 38 21 35 13 4.243 11.397 6.324 17.085 0 40 40 40 4.230 11.462 6.337 17.198 0 40 40 40 4.219 11.168 6.218 16.960 0 31 1 40 4.270 11.573 6.399 17.355 4.263 11.279 6.310 16.818 18 0 0 0 4.270 11.530 6.385 17.293 18 8 10 9 4.274 11.624 6.421 17.439 23 34 34 34 4.253 11.620 6.398 17.459 15 35 24 36 4.251 11.087 6.247 16.499 4.255 11.129 6.262 16.527 22 26 26 22 4.248 11.420 6.335 17.120 15 40 40 40 4.236 11.529 6.359 17.298 9 40 40 40 4.228 11.325 6.279 17.062 7 32 25 36  We now remove the data where ALAE is among the 1% largest, pretending that we have not observed those large values yet. Then we use a Pareto distribution to fit the margins for the rest of the dataset, and use the Gumbel and s.BB2 copulas to fit the dependence structure, respectively. Then simulations based on the two fitted models will be used to assess high senario risks in terms of E[ALAE|LOSS ≥ VaRp (LOSS)]; the results for the Gumbel, s.BB2 copulas, and empirical assessement based on the data will be compared. See Figure 5.3 for a comparison, where calculations were based on the mean of 40 simulations, and 132  for each simulation a random sample of size 106 was used. When the number of simulations was about 40, the mean became stable. First, we fit the two margins separately and get the estimates (Table 5.6). Compared to the estimates for the original data (replicated from Frees and Valdez (1998)), the upper tails of both X1 and X2 become a bit lighter (because of larger ˆ θ). Table 5.6: Estimates for margins with/without 1% largest ALAE removed LOSS(X2 ) ALAE(X1 ) Data Parameters Estimate Standard error Estimate Standard error 1% removed σ 17189 1714 21686 2964 θ 1.310 0.082 3.045 0.323 original σ 14453 1397 15133 1633 θ 1.135 0.066 2.223 0.175  Then we assume that the margins are known as what we have estimated in Table 5.6 and plug them into likelihood functions. Considering there are 31 (originally 34) right-censored loss data (X2 ), the likelihood contributed by a right-censored data value (x1 , x2 ) is f1 (x1 ) − (∂1 F )(x1 , x2 ) = f1 (x1 ) 1 − C2|1 (F2 (x2 )|F1 (x1 )) . The estimation is reported in Table 5.7, and a comparison for different models is illustrated in Figure 5.3. From this example, we can observe that, as the probability level p is very large (eg, 99.5%), “empirical” is not able to extrapolate the increasing trend from the original data and even gives a counter-intuitive decreasing trend, but s.BB2 leads to a reasonable extrapolation beyond the original data although s.BB2 was fitted based on the data with the 1% largest removed. The Gumbel copula seems to be reasonably good as well but has a limited ability in extrapolation compared with s.BB2. In this particular example, s.BB2 outperforms the other models and provides reasonable extrapolation into high-risk scenarios. Furthermore, the choice of δ controls the degree of conservativity of s.BB2.  133  Table 5.7: Comparison between the s.BB2 and Gumbel copulas: from the second order condition of BB2, we know that θ is the second order parameter that dominates the tail behavior of the copula. If we fit δ, θ of s.BB2 simultaneously for the data, then the MLE of δ tends to be very close to 0; that is, s.MTCJ could better fit the data than s.BB2 (since as δ → 0, BB2 becomes MTCJ). But s.MTCJ will lose the asymptotic full dependence structure. The aim of this comparison is not to find the best fitting copula but to study how conservative s.BB2 is. So we fix two values for the parameter δ for s.BB2 (δ = 0.01, 0.1), and then obtain the MLEs of θ. Model s.BB2(δ = 0.01) s.BB2(δ = 0.10) Gumbel  5.6  Parameters θ θ δ  Estimate 0.679 0.520 1.399  Standard error 0.041 0.026 0.028  Blomqvist’s β 0.254 0.226 0.283  Discussion  “Essentially, all models are wrong, but some are useful.” (Box and Draper, 1987). By showing the conservativity of tail comonotonicity and its usage in accounting for unexpected rare events (especially with the promising result illustrated in Figure 5.3), we are not conveying the information that tail comonotonicity is better than other candidate models. Through this study, we hope to be able to raise an issue that has not been attracted enough attention, but may be useful in dependence modeling when one has to assess dependent large risks relying on insufficient information. Since there are very few insurance claim datasets available to the academic researchers and actuaries, more empirical evidence for the usefulness of tail comonotonicity must be welcome. An absolutely continuous copula that is tail comonotonic can provide more realistic conservative bounds on risk measures than the comonotonic copula. Of the simple 2-parameter bivariate copula families, the BB2 family is the most tail asymmetric with lower tail comonotonicity and upper tail independence (upper tail order 2). The BB3 copula is also lower tail comonotonic but has upper tail dependence. As shown in Example 5.2 and Section 5.2.3, BB3 has a slower rate of convergence to the limiting comonotonic tail dependence function. It is shown that the survival 134  Figure 5.3: The value of y-axis is E[ALAE|LOSS ≥ VaRp (LOSS)]. “Empirical” is based on the data with the 1% largest removed; “Empirical0” is based on the original data.  ●  Empirical Empirical0 s.BB2(δ= 0.01) s.BB2(δ= 0.1) Gumbel  120000  100000  ●  ●  ● ●  ● ●  80000 ●  60000 ●  40000  ●  ● ●  ● ● ●  ●  90%  95%  ●  ●  99%  99.5%  p  BB2 copula family provides conservative bounds as a model for bivariate loss data if in fact there is upper tail dependence but not tail comonotonicity. The survival BB3 copula family may also provide bounds if one does not want to use a copula as conservative as BB2. When a tail comonotonic copula is used to provide a data-driven conservative dependence structure, it is suggested to first fit the margins separately then fit the copula based on the uniform scores derived from the fitted margins. If one wants to fit the copula and margins simultaneously, misspecification of the copula model may affect the estimation for margins severely. Moreover, under such a situation, computation may become more complicated. For multivariate loss data, one can use multivariate extensions of the BB2 and  135  BB3 copulas. The BB2 and BB3 families are bivariate Archimedean copulas based on LT families given in Examples 4.1 and 4.2, so there is an obvious extension to exchangeable multivariate Archimedean copulas by the definition. Because the two LT families belong to RV0 , they can be used to get multivariate tail comonotonic copulas with more flexible non-extremal dependence via the mixture of max-id copula families (Joe and Hu, 1996). Although the influence is not large, the choice of the BB2 dependence parameter δ in the data analysis example was arbitrary. The δ can be thought as a parameter to control the degree of conservativity. For future research, we expect that there are certain criteria due to statistical and/or economic reasons to get the parameter δ appropriately tuned. Moreover, we will also look more closely at the tail expansions of E[X1 |X2 > t] and E[X1 |X2 = t] for different marginal distributions of X1 and X2 . The strength of the dependence in the tail can affect whether the conditional tail expectation behaves like O(t), O(tγ ) for some 0 ≤ γ < 1 as t → ∞.  5.7  Proofs  Proof of Proposition 5.1: (a) Write +∞  E[X1 |X2 > t; C ∗ ] =  +∞  P[X1 > x, X2 > t] dx P[X2 > t] 0 1 − F1 (x) − F2 (t) + C ∗ (F1 (x), F2 (t)) dx F 2 (t)  P[X1 > x|X2 > t]dx = 0 +∞  = 0  ∞  = (F 2 (t))−1 E[X1 ] +  {−F2 (t) + C ∗ (F1 (x), F2 (t))}dx .  0  (5.10) Since E[X1 ] < ∞, and P[X1 > x, X2 > t] ≤ P[X1 > x], the integral  ∞ 0 {−F2 (t)+  C ∗ (F1 (x), F2 (t))}dx is finite. The conclusion follows if for all v close to 1 from  136  below, ∞  1  {−v + C ∗ (F1 (x), v)}dx = 0  0 1  ≥ 0  {[−v + C ∗ (u, v)] · [f1 (F1−1 (u))]−1 }du  {[−v + C(u, v)] · [f1 (F1−1 (u))]−1 }du =  ∞  {−v + C(F1 (x), v)}dx. 0  (5.11) The above follows if for any given 0 < u < 1, there exists a v0 = v0 (u) such that 1 ≥ v > v0 implies that C ∗ (u, v) ≥ C(u, v).  (5.12)  Taking a Taylor expansion for g(v) := C ∗ (u, v)−C(u, v) about v with v < v < 1 leads to ∗ C ∗ (u, v) − C(u, v) ∼ (v − v ) C1|2 (u|v )] − C1|2 (u|v )  = (v − v ) C1|2 (u|v )  ∗ (u|v ) C1|2  C1|2 (u|v )  −1 .  ∗ (u|v )/C (u|v ) < 1. (If the limit of the ratio is Then (5.12) holds if limv →1 C1|2 1|2  1, then possibly a non-standard second order expansion is needed.) (b) The copulas for F1 , F2 are C ∗ (u, v) = u + v − 1 + C ∗ (1 − u, 1 − v) and ∗ (u|v) = 1 − C ∗ (1 − u|1 − v) C(u, v) = u + v − 1 + C(1 − u, 1 − v), so that C1|2 1|2 ∗ and C . and C1|2 (u|v) = 1 − C1|2 (1 − u|1 − v). Now part (a) applies to C1|2 1|2  Proof of Corollary 5.2: For any given 0 < u < 1, write ∗ (u|v) C1|2  C1|2 (u|v)  =  ψ (ψ −1 (u) + ψ −1 (v))/ψ (ψ −1 (v)) , ϕ (ϕ−1 (u) + ϕ−1 (v))/ϕ (ϕ−1 (v))  ∗ (u|v)/C (u|v) < 1 if so limv→1− C1|2 1|2  lim  v→1−  ϕ (ϕ−1 (v)) ϕ (ϕ−1 (u)) < , ψ (ψ −1 (v) ψ (ψ −1 (u)  or Υ(u) > Υ(1). The conclusion now follows from Proposition 5.1. 137  Proof of Corollary 5.3: Let ψ = ψδ∗ and ϕ = ϕδ be the LTs for C ∗ and C respectively. By considering the domination term as u → 1− in Table 5.1, Υ(1) = k limu→1− (1 − u)δ  ∗ −δ  = 0 where k is a positive constant. Hence Υ(u) > Υ(1)  for 0 < u < 1 and Corollary 5.2 applies. Proof of Proposition 5.4: Rather than applying part (b) of Proposition 5.1, we provide a more direct proof. Because we are using survival copulas, rather than (5.12), we want to show that for any given 0 < u < 1, there is a v0 such that C ∗ (u, v) ≥ C(u, v) or C ∗ (1 − u, 1 − v) = Cψ (1 − u, 1 − v) ≥ Cϕ (1 − u, 1 − v) = C(1 − u, 1 − v) for 1 ≥ v > v0 . The inequality is the same as ψ(ψ −1 (u1 ) + ψ −1 (v1 )) ≥ ϕ(ϕ−1 (u1 ) + ϕ−1 (v1 )),  (5.13)  with u1 = 1 − u and v1 = 1 − v for v1 near 0. Let u1 = ϕ(su ) and v1 = ϕ(sv ). Since ψ is decreasing, inequality (5.13) is the same as ψ −1 (u1 ) + ψ −1 (v1 ) ≤ (ψ −1 ◦ ϕ)(ϕ−1 (u1 ) + ϕ−1 (v1 )) or (ψ −1 ◦ ϕ)(su ) + (ψ −1 ◦ ϕ)(sv ) ≤ (ψ −1 ◦ ϕ)(su + sv ). For s sufficiently large, if ξ(s) := ψ −1 ◦ϕ(s) is strictly convex, then for any w > 0, ξ(s+w)−ξ(s) is strictly increasing in s with limit of ∞, since ξ(s) is also a strictly increasing function. Therefore, for sv large enough, ξ(su + sv ) − ξ(sv ) > ξ(su ). If Υ(u) is strictly decreasing in u near 0, that is, ξ (s) = ϕ (s)/ψ (ξ(s)) is strictly increasing for large s by letting s = ϕ−1 (u), then ξ(s) is strictly convex for large s, which completes the proof. Proof of Corollary 5.7: Since 1 − ψ(s) = increasing in t as t →  0+ ,  s 0 −ψ  (t)dt and −ψ (t) is strictly  by Lemma 5.6, −ψ (s)/(1 − ψ(s)) = o(s−1 ), s → 0+ .  Therefore, by L’Hˆopital’s rule, lim  s→0+  log(1 − ψ(s)) [log(1 − ψ(s))] ψ (s)/[1 − ψ(s)] = lim = lim = 0, + + log(1 − ϕ(s)) s→0 [log(1 − ϕ(s))] s→0 ϕ (s)/[1 − ϕ(s)]  which implies ϕ(s) approaches 1 faster than ψ(s) as s → 0+ , and lims→0+ [1 − 138  ψ(s)]/[1 − ϕ(s)] = ∞. Thus, limu→1− ψ −1 (u)/ϕ−1 (u) = 0, and lim Υ(u) = lim  u→1−  u→1−  ϕ (ϕ−1 (u)) ψ −1 (u) [ψ −1 (u)] = lim = lim = 0. ψ (ψ −1 (u)) u→1− [ϕ−1 (u)] u→1− ϕ−1 (u)  Then Corollary 5.2 implies the result. Proof of Corollary 5.8: Since ψ(s) =  ∞ s −ψ  (t)dt and −ψ (t) is strictly decreas-  ing in t as t → ∞, by Lemma 5.5, −ψ (s)/ψ(s) = o(s−1 ), s → ∞. Therefore, by L’Hˆopital’s rule, log ψ(s) [log ψ(s)] ψ (s)/ψ(s) = lim = lim = 0, s→∞ log ϕ(s) s→∞ [log ϕ(s)] s→∞ ϕ (s)/ϕ(s) lim  which implies that ϕ(s) goes to 0 faster than ψ(s) as s → ∞ and lims→∞ ϕ(s)/ψ(s) = 0. Thus, limu→0+ ϕ−1 (u)/ψ −1 (u) = 0. Then write 1 ψ (ψ −1 (u)) −ψ (ψ −1 (u))ψ −1 (u) ϕ(ϕ−1 (u)) ϕ−1 (u) := = × × . Υ(u) ϕ (ϕ−1 (u)) ψ(ψ −1 (u)) −ϕ (ϕ−1 (u))ϕ−1 (u) ψ −1 (u) So there exists a 0 < K < ∞, such that 0 ≤ lim  u→0+  1 −sψ (s) ϕ−1 (u) ≤ lim K × lim −1 = 0. Υ(u) s→∞ ψ(s) u→0+ ψ (u)  Then Proposition 5.4 implies the result. Proof of Proposition 5.9: We prove only part (a), since part (b) follows from (a) as in Proposition 5.1. Since E[X1 ] is finite, E[X1 |X2 ] is finite a.e. F2 . Write E[X1 |X2 = t; C ∗ ] +∞  =  +∞  f2 (t) − (∂2 F )(x, t) dx f2 (t) 0 ∗ (F (x)|F (t))f (t) +∞ f2 (t) − C1|2 1 2 2 ∗ dx = {1 − C1|2 (F1 (x)|F2 (t))}dx, f2 (t) 0  P[X1 > x|X2 = t]dx = 0 +∞  = 0  139  The conclusion follows if with v = F2 (t) < 1, ∞ 0  ∗ (F1 (x)|v)}dx {C1|2 (F1 (x)|v) − C1|2 ∞  C1|2 (F1 (x)|v) 1 −  = 0  ∗ (F (x)|v) C1|2 1  C1|2 (F1 (x)|v)  dx ≥ 0,  (5.14)  as t → ∞. A sufficient condition is lim  v→1−  ∗ (u|v) C1|2  C1|2 (u|v)  < 1,  0 < u < 1.  Proof of Corollary 5.10: From the assumption, ∗ (u|v) C1|2  lim  v→1−  C1|2 (u|v)  = lim  v→1−  ψ ψ −1 (u) + ψ −1 (v) /ψ ψ −1 (v) limv→1− Υ(v) <1 = −1 −1 −1 Υ(u) ϕ ϕ (u) + ϕ (v) /ϕ ϕ (v)  so that Proposition 5.9 applies. Proof of Corollary 5.11: From part (b) of Proposition 5.9, we want to show that for ∗ (u|v)]/[1 − C (u|v)] < 1, or fixed 0 < u < 1, limv→0+ [1 − C1|2 1|2 ∗ lim C1|2 (u|v)/C1|2 (u|v) > 1,  v→0+  i.e., lim  v→0+  ϕ ϕ−1 (u) + ϕ−1 (v) ψ ψ −1 (u) + ψ −1 (v) =: g(0) > h(0) := lim . ψ (ψ −1 (v)) ϕ (ϕ−1 (v)) v→0+ (5.15)  The strict convexity condition on ψ −1 ◦ ϕ and the proof of Proposition 5.4 imply that with u fixed, ψ(ψ −1 (u) + ψ −1 (v)) =: G(v) > H(v) := ϕ(ϕ−1 (u) + ϕ−1 (v)),  (5.16)  for v > 0 sufficiently small. Also, G(0) = H(0) = 0, so we must have G (0) > 140  H (0) for any 0 < u < 1, thus (5.15) holds. since G = g and H = h. Proof of Corollary 5.12: The proof is straightforward due to Corollary 5.10 and the proof of Corollary 5.7. Proof of Corollary 5.13: The proof is straightforward due to Corollary 5.11 and the proof of Corollary 5.8. Proof of Lemma 5.14: It can be written that       C(uw1 , . . . , uwd ) = ψ ψ −1 (uw∗ ) 1 + i∈{j:wj =w∗ }  ψ −1 (uwi )  . ψ −1 (uw∗ )  Letting ti (u) := ψ −1 (uwi )/ψ −1 (uw∗ ), since ψ ∈ RV0 , ψ −1 ∈ RV∞ (0+ ), and thus wi > w∗ implies that limu→0+ ti (u) = 0 for any i ∈ {j : wj = w∗ }. Let t∗ := (ti )i∈{j:wj =w∗ } , and define h(t∗ ) := ψ(s(1 + expansion for  h(t∗ )  i∈{j:wj =w∗ } ti )).  A Taylor  about 0 leads that h(t∗ ) ∼ ψ(s) + sψ (s) ×  ti . i∈{j:wj  (5.17)  =w∗ }  Then choosing s = ψ −1 (uw∗ ) in (5.17) proves the claim, and (u, w) = o(u), u → 0+ is due to Lemma 5.5. Proof of Proposition 5.15: It is well known that CTE is subadditive for continuous risks (Denuit et al., 2005); that is, for any X ∈ R(F1 , . . . , Fd ) d  CTEp (X1 + · · · + Xd ) ≤  CTEp (Xi ),  0 < p < 1.  i=1  From Proposition 4.15, for such an X, d  CTEp (X1 + · · · + Xd ) ∼  d  CTEp (Xi ) = i=1  CTEp (Xi ), i=1  141  p → 1− .  Thus, CTEp (X1 + · · · + Xd )  CTEp (X1 + · · · + Xd ),  p → 1− .  Proof of Proposition 5.17: By Theorem 4.3 of Embrechts et al. (2009a), VaRp is asymptotically subadditive for X as p → 1− . Also, X satisfies Assumption 5.1, and thus VaRp is asymptotically additive for X. So the above asymptotic inequality holds, and tail comonotonicity becomes an asymptotically worst dependence structure in this sense. Proof of Proposition 5.18: By Theorem 6.8 of McNeil et al. (2005), VaRp is subadditive for X when 0.5 ≤ p < 1. Also, notice that if the assumption that X is non-negative in Proposition 4.14 is relaxed, the asymptotic additivity of VaR still holds. Note that the X here is not elliptical; within elliptical distribution families, there are not non-trivial tail comonotonic distributions (see Chapter 4).  142  Chapter 6  Second order regular variation and risk measures 6.1  Introduction  In actuarial science, some statistical quantities about a random variable X are often referred to as risk measures. Among many of them, value at risk (VaR) and conditional tail expectation (CTE) are the most popular risk measures. For 0 < p < 1 and usually with p > 0.9, VaRp refers to the 100p percentile of loss and CTEp is a conditional expectation given the loss exceeds VaRp . Both of them have been adopted by regulations of insurers. For example, CTE has been required for calculating the relevant risks of segregated fund in Canada (OSFI, 2011, Chapter 8). VaR has been commonly-used for financial and insurance risks while recently CTE has been suggested to be a more conservative risk measure compared to VaR, especially when loss distributions are heavy-tailed. We refer to Denuit et al. (2005) and McNeil et al. (2005) for relevant risk measures and their relationships. More precisely, the class of heavy-tailed distributions is a very large family and can be defined as K = {F df on (0, ∞) :  +∞ x e dF (x) 0  = ∞ for all  > 0} (See Fig-  ure 1.4.1 of Embrechts et al. (1997) for the classes of heavy-tailed distributions). In this chapter, we only study an important subclass of the heavy-tailed distributions, consisting of distributions that are supported on the positive real line and have regularly varying survival functions. For narrative convenience, we use the 143  term “heavy-tailed distribution” for this subclass of heavy-tailed distributions. From the viewpoint of risk management, tail behavior of loss random variables is important. In the literature, much work has been done to better understand the extremal patterns of aggregate losses by doing asymptotic analysis, such as, Alink et al. (2004, 2005, 2007); Barbe et al. (2006); Embrechts et al. (2009b). In these papers, asymptotic behavior of aggregate dependent losses has been studied. We believe that the asymptotic study on the tail behavior is relevant to understand the magnitude of large losses and the performance of corresponding risk measures. The benefit of doing asymptotic analysis on risk measures, on the other hand, may establish asymptotic relationships between risk measures. For example, as a coherent risk measure, CTE has many advantages over VaR; we refer to Artzner et al. (1999) for limitations of VaR compared with a coherent risk measure. However, computing CTE is often more costly than estimating VaR since the former often involves Monte Carlo simulations for non-closed forms of integrations. To this end, studying asymptotic relationships between CTE and VaR becomes promising. Moreover, by studying the asymptotic relationship between CTE and VaR, we will be able to have better sense of how VaR may underestimate heavy-tailed risks, and we refer to Figure 6.1 for an example. Some asymptotic analysis of CTE in terms of VaR has been done most recently by Zhu and Li (2012). In their paper, the non-negative random vector under study has the form of X = (T1 R, . . . , Td R), where R is non-negative, independent of T = (T1 , . . . , Td ) ≥ 0 and has a regularly varying survival function. Zhu and Li (2012) derived the closed-form first order approximation for CTE of the following form: E[X1 | X > VaRp ( X )], as p → 1. Their result is interesting since the closed-form approximation of CTE only relies on some mild moment conditions on Ti ’s. The results obtained in Zhu and Li (2012) is only a first order approximation of the relationship between CTE and VaR, as p → 1. The first order property is useful since we have known more about VaR while the asymptotic relationship will provide some insights about CTE. However, the limitation is also obvious that the first order asymptotic approximation may not be good for a sub-extremal threshold. So, second order properties would be able to provide more insightful information. On one hand, the second order properties determine the rates of convergence for the first order approximation, and when the speed of convergence is higher, the 144  asymptotic approximation by only the first order term has a better chance to provide a good approximation for the sub-extremal level; on the other hand, the second order term itself improves the approximation. In the asymptotic analysis of CTE, second order regular variation (2RV) provides a tractable tool for studying second order properties. 2RV was originally studied in the extreme value theory, to study the speed of convergence of the extreme value condition (de Haan and Ferreira, 2006, Section 2). For a general theory of 2RV, we refer to de Haan and Ferreira (2006); de Haan and Stadtm¨uller (1996) for references. In this chapter, we use 2RV to study the tail behavior of the distribution function F (x), which is more naturally related to the study of risk measures. In Geluk et al. (1997), 2RV on survival functions was studied in the context of convolution and so on. We refer to Degen et al. (2010) for a recent study on risk concentration and diversification under the assumption of 2RV, which studied risk concentration and diversification benefit of multiple losses. In general, we are interested in the behavior of E[Xi |g(X1 , . . . , Xd ) > t],  t → ∞,  (6.1)  where g is a loss aggregate function, and (X1 , . . . , Xd ) has a general dependence structure and the margins have some particular tail patterns. However, the scale mixture approach studied in Zhu and Li (2012) provides a wide subset of such random vector X. We will follow this approach in this chapter and study the second order property of (6.1) for the case where g is a homogeneous function of order 1 and the non-negative X is constructed by the scale mixture approach. More specifically, we will study the rates of convergence of asymptotic relationships between CTE and VaR under the framework of 2RV. Closed-form second order approximations have been obtained for both the univariate case and the multivariate case where the random vector is constructed from a scale mixture with a heavy-tailed non-negative random variable. For both cases, we find that the first order approximation is affected by only the regular variation index −α of marginal survival functions, while the second order approximation is influenced by both the parameters for first and second order regular variation, and the convergence speed is dominated by the second order parameter only. Many well-known continuous dis145  tributions for modeling univariate/multivariate losses or financial returns — such as the Student tν , Burr, multivariate tν , multivariate distributions constructed by regularly varying margins and some upper tail dependent copulas, and some elliptical distributions with regularly varying margins — satisfy the second order conditions that we will study in this chapter. Some side results are second order properties for the well-known Karamata’s Theorem and Breiman’s Theorem. This chapter is organized as following: In Section 6.2, notation, preliminary concepts and results are presented. In Section 6.3, the univariate case with 2RV conditions is studied to understand the asymptotic relationship between CTE and VaR. The study on the multivariate case is reported in Section 6.4. Some discussion of future research is in Section 6.5. Finally, Section 6.6 presents all the proofs.  6.2  Preliminaries  Fundamental knowledge about regular variation has been introduced in Section 2.3. For the multivariate case to be studied in Section 6.4, we will study positive random vector X constructed as this form (X1 , . . . , Xd ) = (RT1 , . . . , RTd ), where the tail behavior of product of two random variables plays an important role. The following result just tells us how the product will inherit the tail behavior of the random variable that has a heavier tail. Proposition 6.1 (Breiman’s Theorem) Suppose X is a non-negative random variable such that its survival function F (x) ∈ RV−α with α > 0, and Y ≥ 0 is a random variable, independent of X, with E[Y α+ ] < ∞ for some > 0. Then P[XY > x] ∼ E[Y α ]P[X > x],  x → ∞.  (6.2)  That is, if X is regularly varying, then its right tail behavior will be inherited by the product XY , where Y has a lighter right tail and a finite moment of order higher than E[Y α ]. We refer to Breiman (1965) for a proof of the case where 0 < α < 1, and the proof is also adaptable for proving the case where 0 < α. For some multivariate versions of Breiman’s Theorem, we refer to Resnick (2007). The following definition of the second order regular variation comes from de Haan and Stadtm¨uller (1996), de Haan and Ferreira (2006) and Geluk et al. 146  (1997). See also Neves (2009) for a slightly different version of extended second order regular variation and Wang and Cheng (2005) for third and higher order regular variation. Definition 6.1 If the survival function of a non-negative random variable X is F := 1 − F and F : [0, ∞) → (0, 1] satisfies that F ∈ RV−α with α > 0. Then F is said to be of second-order regular variation with parameter ρ ≤ 0, if there exists a function A(t) that ultimately has a constant sign with limt→∞ A(t) = 0 and a constant c = 0 such that F (tx)/F (t) − x−α = Hα,ρ (x) = cx−α t→∞ A(t)  x  uρ−1 du,  lim  x > 0.  (6.3)  1  Then it is written as F ∈ 2RV−α,ρ and A(t) is referred to as the auxiliary function of F . It is known from de Haan and Stadtm¨uller (1996) or a more relevant form in Geluk et al. (1997) that if Hα,ρ (x) is not a multiple of x−α then ρ < 0 implies that there exists a c = 0 such that Hα,ρ (x) = cx−α  xρ − 1 , ρ  |A| ∈ RVρ and no other choices of ρ are consistent with A(t) → 0. Moreover, convergence in (6.3) is uniform in x on compact intervals of (0, ∞). By adjusting A(t), we can always let c = 1, and unless otherwise specified, c is assumed to be 1. We only study the case where ρ < 0 and α > 1 in this chapter. Also, note that for second order regular variation, ρ = −∞ (Resnick, 2007, page 68). Therefore, the Pareto distribution with the cdf F (x) = 1 − (k/x)α , x ≥ k, where Hα,ρ (x) in (6.3) would be 0 for any function A(t), does not satisfy the conditions of second order regular variation and thus will not be studied in this chapter. To illustrate the main result for the multivariate case, we will use the concept of Archimedean copula as a mixture.  147  6.3  Univariate cases  For the univariate case, a direct application of Karamata’s Theorem (Theorem 2.2) shows that, for a non-negative random variable X ∈ RV−α with α > 1, CTEp (X) is finite and CTEp (X) ∼  α VaRp (X), α−1  p → 1.  (6.4)  From (6.4), we know that when p is sufficiently large, CTE and VaR of a regularly varying random variable have a deterministic relationship that is determined by the regular variation index −α. However, we do not know how close is the right-hand side of (6.4) to its left-hand side. In practice, we always choose a level p that is strictly less than 1 to evaluate risks. To this end, a higher order approximation is useful. A closed-form second order approximation is ideal. If one can not obtain the closed-form second order approximation, knowledge about how fast the righthand side converges to the left-hand size of (6.4) is also informative. In this section, we will derive the closed-form second order approximation for the univariate case assuming that the loss random variable is not only regularly varying with index −α, but also second regularly varying. This assumption is mild, since some useful parametric distributions, such as Student tν , satisfy it (see Example 6.3). Moreover, as a technical tool for our proof, Proposition 6.5 at the end of this section shows some interesting second order properties with respect to Karamata’s Theorem. Now, we are ready to give a technical lemma showing that the slowly varying function involved in the second order regular variation has a certain nice form. Lemma 6.2 Suppose F ∈ 2RV−α,ρ , α > 0, ρ < 0, then we can write F (t) = kt−a (t), t > 0 with some k > 0 and limt→∞ (t) = 1, and moreover we have |1 − (t)| ∈ RVρ . In what follows, we give a result of a uniform inequality for the second order regular varying survival distribution function. This result is essentially Theorem 2.3.9 of de Haan and Ferreira (2006) and will be useful to prove our subsequent propositions. We refer to de Haan and Ferreira (2006) and Cheng and Jiang (2001) for relevant discussion.  148  Proposition 6.3 Suppose F ∈ 2RV−α,ρ with α > 0 and ρ < 0, then for any , δ > 0, there exists t0 = t0 ( , δ) such that for all t, tx ≥ t0 and x > 0, F (tx)/F (t) − x−α xρ − 1 ≤ max x−α+ρ+δ , x−α+ρ−δ , − x−α a(t) ρ  (6.5)  where a(t) = −ρ[1 − (t)]/ (t) with F (t) = kt−α (t), 0 < k < +∞ and limt→∞ (t) = 1. Remark 6.1 From Proposition 6.3, we may choose the A(t) in Definition 6.1 as A(t) := a(t) = −ρ[1 − (t)]/ (t), and clearly |A(t)| ∈ RVρ . The following result tells us that the rates of approximation of CTEp (X) to α α−1 VaRp (X) are determined by the second order parameter ρ and are of the order of [VaRp (X)]ρ+1 . When ρ is smaller (more negative), the speed of convergence is faster. In other words, when ρ is more negative, the first order term can provide a better approximation. Proposition 6.4 Suppose that a non-negative random variable X ∈ 2RV−α,ρ with α > 1 and ρ < 0, and write its survival function F (t) := kt−α (t). Then CTEp (X) ∼  α VaRp (X) + η(VaRp (X)), α−1  p → 1,  (6.6)  where η(t) =  ta(t) , (α − 1 − ρ)(α − 1)  (6.7)  and a(t) can be chosen as a(t) := −ρ[1 − (t)]/ (t). Remark 6.2 Since |a(t)| ∈ RVρ and thus |ta(t)| ∈ RVρ+1 , Equation (6.7) just shows that the second order term becomes more important as ρ > −1 (because |ta(t)| is increasing as t → ∞). So we comment that the second order approximation is especially useful for the case where −1 < ρ < 0, in which the second order term may not be ignored. In Example 6.5, we will illustrate by an example for a multivariate case that the second order term significantly improves the approximation. 149  Remark 6.3 Proposition 6.4 just indicates that the first order term depends on α only, and the second order term depends on both α and ρ. The proof in Proposition 6.4 also shows that for α > 1 and ρ < 0, as t → ∞, verges to  ρ x−α x ρ−1  F (tx)/F (t)−x−α A(t)  con-  uniformly in x on any set of the form [x0 , ∞), where x0 > 0.  Then we can exchange the orders of limit and integral operations in order to get limt→∞  ∞ P[X>ts]/P[X>t]−s−α ds a A(t)  for some a > 0.  Remark 6.4 A necessary condition for the asymptotic relationship between CTE and VaR is that, the random variable can not be too heavy-tailed; that is, the regular variation index α should be greater than 1. When α ≤ 1, the tail behavior becomes much more complicated. In particular, as α < 1, CTEp (X) = ∞, ∀p < 1, and under the current framework, one cannot study the asymptotic relationship between CTE and VaR. Example 6.1 (Hall/Weiss class) Suppose X is a random variable with support [0, ∞) and distribution function F such that with x ≥ 1, α > 1, ρ < 0, F (x) = 1 −α (1 2x  + xρ ). Then clearly, F ∈ 2RV−α,ρ , and when t := VaRp (X) is big  enough, CTEp (X) = t +  ∞ 1 −α (1 + xρ )dx t 2x 1 −α (1 + tρ ) 2t  =t+  t/(α − 1) + tρ+1 /(α − 1 − ρ) , 1 + tρ  which implies, with (1 + tρ )−1 ∼ 1 − tρ , t → +∞, CTEp (X) −  α ρ VaRp (X) ∼ [VaRp (X)]ρ+1 , α−1 (α − 1 − ρ) × (α − 1)  p → 1.  It means that the rate of convergence is determined by the second order parameter ρ and is of the order of [VaRp (X)]ρ+1 . Figure 6.1 illustrates the effects of ρ. Example 6.2 (Burr distribution) Consider the Burr distribution with the survival function F (x) = (1 + xb )−a , x, a, b > 0, and the second order expansion leads to F (x) = x−ab [1 − ax−b + o(x−b )], so that F ∈ 2RV−ab,−b . 150  x → ∞,  Figure 6.1: Sub-extremal relationships between CTE and VaR (α = 2) for Hall/Weiss class Effect of the second order parameter ρ  2.00  ρ=−2 ρ=−1 ρ=−0.5  CTE / VaR  1.98 1.96 1.94 1.92 1.90  0.95  0.96  0.97  0.98  0.99  1.00  Probability level  Now we evaluate how large p must be for the second order approximation of Proposition 6.4 to be decent for the Burr distribution. Assume X ∼ Burr(a, b, σ), where a, b are the two shape parameters and σ is the scale parameter; that is, X ∼ F (x) := 1 − 1 + (x/σ)b  −a  . Write F (x) = (1 + (x/σ)b )−a = σ ab x−ab (x),  where (t) = ((t/σ)−b + 1)−a is a slowly varying function. According to Proposition 6.4, an auxiliary function for the Burr distribution can be chosen as A(t) := b[1 − (t)]/ (t) = b  151  1 + (t/σ)−b  a  −1 .  Clearly, A ∈ RV−b . Then, letting t := VaRp (X), CTEp (X) = t + (1 − p)−1  ∞  1 + (x/σ)b  −a  dx  t (1+(t/σ)b )−1  σ (1 − y)1/b−1 y a−1/b−1 dy b(1 − p) 0 σ =t+ Beta (1 + (t/σ)b )−1 ; a − 1/b, 1/b , b(1 − p)  =t+  (6.8)  where Beta(·; a − 1/b, 1/b) is an incomplete Beta function with parameters a − 1/b, 1/b. We can use (6.8) to get the exact calculation, and then compare it to the second order approximation. Table 6.1 presents the values of p for which a decent second order approximation can be obtained. For a given leading parameter (ab) for the tail heaviness, a larger b leads to a smaller p; that is, the second order parameter (ρ = −b) is more negative and thus the rates of convergence becomes faster. More interestingly, for a given b, the second order approximation for a lighter tail (larger α) is not as good as that for a heavier tail. When the tail becomes very light (say, α ≥ 4), it was hard to get a good second order approximation within the 5% error band. The smallest p’s required that are reported in Table 6.1 do not rely on the scale parameter. However, theoretically, there are infinitely many choices for the auxiliary function A(t). The smallest p required to get a decent second order approximation heavily rely on the choice of A(t). Table 6.1: How large p must be to get a good second order approximation: the values are the corresponding p for which the absolute difference between the second order approximation and the true value is 5% of the true value.  b 0.5 1 1.5  1.1 0.86977 0.32947 0.11240  α(= ab) 1.5 2 0.98828 0.99859 0.73711 0.88890 0.44426 0.65444  3 0.99994 0.97825 0.86267  The next result extends Karamata’s theorem to a second order regular condition for the case where regular variation index −α < −1. Since some continu152  ous random variables, such as the Student tν random variable, have closed-form density functions but do not have closed-form cdf’s, the following result is very useful to derive second order properties, such as the second order parameter and the corresponding auxiliary function, for their survival functions from their density functions. Proposition 6.5 Let g ∈ 2RV−α,ρ , α > 1, ρ < 0, with an auxiliary function A(x), ∞ ∗ t g(x)dx. Then g α−1 A∗ (t) = α−1−ρ A(t), and  and define g ∗ (t) := iary function is g ∗ (t) ∼  t tA(t) g(t) + α−1 ρ  ∈ 2RV−α+1,ρ , the corresponding auxil-  1 1 − α−ρ−1 α−1  g(t),  t → ∞.  (6.9)  Example 6.3 (Student tν distribution) Consider the standard Student tν distribution with density function Γ((ν + 1)/2) f (x) = √ νπΓ(ν/2)  1+  x2 ν  −(ν+1)/2  ,  ν > 1.  Then it can be verified that Γ((ν + 1)/2) (ν+1)/2 −ν−1 ν x 1 − 2−1 ν(ν + 1)x−2 , f (x) ∼ √ νπΓ(ν/2)  x → ∞.  That is, f (x) ∈ 2RV−ν−1,−2 . By Proposition 6.5, its survival function F ∈ 2RV−ν,−2 . For the Student tν distribution, the second order parameter is always that ρ = −2. The second order property can also be obtained from scale mixture representation of the tν distribution. In Example 6.4 in the next section, we will derive the same thing from a mixture with a generalized inverse Gamma distribution.  6.4  Multivariate cases  From the actuarial viewpoint, we are interested in CTEs of the form E[X1 |g(X1 , ..., Xd ) > t], and in particular E[X1 |X2 > t] for the bivariate case, where t is large, and X1 , ..., Xd possess some general dependence structure and are heavy-tailed loss 153  random variables (with the same RV index). In Chapter 3, the concept of tail order has been proposed to study the degree of positive tail association that covers asymptotic dependence and asymptotic independence. In Chapter 3, the upper tail order of a copula C is denoted as κU (C). And κU (C) = 1 corresponds to the usual upper tail dependence case and κU (C) > 1 covers asymptotic independence and even negative dependence cases. For E(X1 |X2 > t), some examples and rough analysis suggests that under some regularity conditions on the copula C of (X1 , X2 ), κU (C) = 1 ⇒ E[X1 |X2 > t] = O(t), κU (C) > 1 ⇒ E[X1 |X2 > t] = o(t),  t → ∞; t → ∞.  However, it is difficult to find sufficient conditions to deduce general results for the order of t and the leading coefficient. In Chapter 7, we have more discussions on the tail behavior of the above conditional specifications. The study of second order terms would depend on developing a theory for the second order tail functions of copulas (Joe and Li, 2011). A special example with multivariate regular variation and upper tail dependence is the case of heavy-tailed scale mixtures studied by Zhu and Li (2012). They obtained E[X1 |X2 > t] ∼ at, t → ∞, with a being a constant relying on some finite moments. In this section, one of our main results is to assume 2RV and get an extension to Zhu and Li’s result to a second order approximation. We also show that heavy-tailed scale mixtures with 2RV conditions covers some parametric distribution families that are commonly used in actuarial science; that is, it is not too restrictive for applications. Now, consider bivariate cases first. Let (X1 , X2 ) = (RT1 , RT2 ), where (T1 , T2 ) ≥ 0, independent of R ≥ 0. Note here we do not specify the dependence structure between T1 and T2 . This class of distributions covers multivariate Pareto distributions and truncated multivariate elliptical distributions, etc. With some regularity conditions on R and moment conditions on T1 and T2 , the  154  following asymptotic relationship has been derived in Zhu and Li (2012), E[X1 |X2 > VaRp (X2 )] ∼  α E[T1 T2α−1 ] VaRp (X2 ), α − 1 E[T2α ]  p → 1.  (6.10)  In this section, we obtain the rate of the convergence of (6.10) and the second order approximation. Proposition 6.6 Let (X1 , X2 ) = (RT1 , RT2 ), where (T1 , T2 ) ≥ 0, independent of R ≥ 0 such that the survival function of R is F R (t) ∈ 2RV−α,ρ with α > 1 and ρ < 0. If E[T1 ] < ∞, E[T1 T2α−1−ρ+ξ ] < ∞ for some ξ > 0 and E[T2α−ρ+δ ] < ∞ for some δ > 0, then an extension of (6.10) can be represented by E[X1 |X2 > VaRp (X2 )] ∼ η1 (VaRp (X2 )) + η2 (VaRp (X2 )),  p → 1,  where η1 (t) =  α E[T1 T2α−1 ] t; α − 1 E[T2α ]  η2 (t) =  A(t)t ρ  α − ρ E[T1 T2α−1−ρ ] α E[T1 T2α−1 ]E[T2α−ρ ] − α−1−ρ E[T2α ] α−1 (E[T2α ])2  .  Here, A(t) is the auxiliary function for F R (t). Remark 6.5 Similarly to the univariate case, the first order term only depends on α, and the second order term depends on α and ρ. In order to get a higher order approximation, existence of higher moments of the form E[T1 T2α−1−ρ+ξ ] for some ξ > 0 and E[T2α−ρ+δ ] < ∞ for some δ > 0 are necessary. The speed of convergence is dominated by the second order parameter ρ only. Since |A(t)| ∈ RVρ , the convergence speed of E[X1 |X2 > VaRp (X2 )] → of the order of [VaRp (X2  )]ρ+1 ;  α−1 ] α E[T1 T2 α−1 E[T2α ] VaRp (X2 )  is  that is, when ρ is more negative, the convergence  is faster at the sub-extremal level. It is also consistent with the univariate case. In Proposition 6.6, we do not specify the dependence structure between T1 and T2 . If we have a random vector T ≥ 0 and the conditioning event can be of the form h(X) > t where h : Rd+ → R+ is homogeneous of order 1, then h(X) = h(T)R and we can take T2 := h(T). 155  Proposition 6.7 Suppose there is a function h : Rd+ → R+ , homogeneous of order 1, and a random vector X := (RT1 , . . . , RTd ) where T = (T1 , . . . , Td ) ≥ 0 independent of R ≥ 0 and F R ∈ 2RV−α,ρ with α > 1 and ρ < 0. If E[T1 ] < ∞, E[T1 hα−1−ρ+ξ (T)] < ∞ for some ξ > 0 and E[hα−ρ+δ (T)] < ∞ for some δ > 0, then E[X1 |h(X) > VaRp (h(X))] ∼ η1 (VaRp (h(X))) + η2 (VaRp (h(X))),  p → 1,  where α E[T1 hα−1 (T)] t; α − 1 E[hα (T)] α − ρ E[T1 hα−1−ρ (T)] α E[T1 hα−1 (T)]E[hα−ρ (T)] A(t)t − η2 (t) = ρ α−1−ρ E[hα (T)] α−1 (E[hα (T)])2 η1 (t) =  Remark 6.6 Examples of the homogeneous h function are norms of vectors, such as the l1 -norm and l∞ -norm. These two norms correspond to the CTEs of a marginal risk X1 respectively conditioning on the events that the sum of all marginal risks and the maximum of all marginal risks are higher than a threshold. Example 6.4 A d-dimensional elliptical distribution can be constructed by X = (X1 , . . . , Xd ) = RD(U1 , . . . , Um ), where (U1 , . . . , Um ) is uniformly distributed on the surface of the unit hypersphere in Rm , D is a d × m matrix with DDT positive semi-definite, and R is a positive random variable with F R ∈ 2RV−α,ρ and the auxiliary function is A(x). We may take T = (T1 , . . . , Td ) = D(U1 , . . . , Um ), and Ti+ = 0 ∨ Ti for i = 1, . . . , d, then the form X+ = RT+ fits into the study of this chapter. X+ follows a left-truncated  1  elliptical distribution. Taking the  homogeneous function h in Proposition 6.7 to be the l1 norm, and letting t := VaRp  d + i=1 Xi  , then we have for any k ∈ {1, . . . , d}, d  E  Xk+  Xi+ > t ∼ η1 (t) + η2 (t),  p → 1,  i=1 1 Unless otherwise stated, left-truncated in this chapter refers to left-truncating below zeros for each element of a random vector.  156  .  where η1 (t) =  α E[Tk ( α − 1 E[(  η2 (t) =  A(t)t ρ  −  d α−1 ] i=1 Ti ) t; d α i=1 Ti ) ]  α − ρ E[Tk ( α−ρ−1 E[(  α E[Tk ( α−1  d α−1−ρ ] i=1 Ti ) d α i=1 Ti ) ]  d α−1 ]E[( d T )α−ρ ] i=1 i i=1 Ti ) d α (E[( i=1 Ti ) ])2  .  In particular, we may choose a bivariate standard normal random vector (T1 , T2 ) with identical margins, correlation 0 <  < 1, and let Ti+ = 0 ∨ Ti for i = 1, 2,  and R = Y −1/2 with Y following Gamma(ν/2, ν/2) in which ν/2 are both shape and rate parameters. Then the scale mixture (X1 , X2 ) = (RT1 , RT2 ) is a bivariate tν distribution, and (X1+ , X2+ ) = (RT1+ , RT2+ ) is a left-truncated bivariate tν distribution. Note that Y −1 follows an Inverse Gamma distribution with both the shape and scale parameters being ν/2. Thus the density function for Y −1 is fY −1 (x) =  (ν/2)ν/2 −ν/2−1 −x−1 ν/2 x e , Γ(ν/2)  x > 0; ν > 0.  Therefore, the density function of R can be derived as fR (x) = 2fY −1 (x2 )x = 2 ∼2  (ν/2)ν/2 −ν−1 −x−2 ν/2 x e Γ(ν/2) (ν/2)ν/2 −ν−1 x (1 − x−2 ν/2), x → ∞. Γ(ν/2)  Then it can be verified that fR (x) ∈ 2RV−ν−1,−2 and the corresponding auxiliary function AfR (x) = νx−2 . Hence F R (x) ∈ 2RV−ν,−2 by Proposition 6.5 and the corresponding auxiliary function AF R (x) =  (ν+1)−1 (ν+1)−1−(−2) AfR (x)  =  ν 2 −2 ν+2 x .  Moreover, from (6.25), we know that the product Xi = RTi will inherit the right tail behavior of R; that is, the univariate Student tν random variable Xi ∈ 2RV−ν,−2 , which is also consistent to Example 6.3, where the same conclusion is drawn directly from the density function of the tν distribution. Then more specifically, letting ν = 2 and t := VaRp (X1+ + X2+ ), then for the bivariate left-truncated 157  tν distribution, E X1+ X1+ + X2+ > t ∼ η1 (t) + η2 (t),  p → 1,  where η1 (t) = 2  E[T1+2 ] + E[T1+ T2+ ] t; E[(T1+ + T2+ )2 ]  η2 (t) =  −  2 E[T1+ (T1+ + T2+ )3 ] E[T1+2 + T1+ T2+ ]E[(T1+ + T2+ )4 ] + 3 E[(T1+ + T2+ )2 ] (E[(T1+ + T2+ )2 ])2  AF R (t)t.  The higher moments of truncated standard bivariate tν random variables can be calculated in what follows by some recurrence relations in Shah and Parikh (1964) or page 313–314 of Kotz et al. (2000). That is, η1 (t) = t; η2 (t) = g( )AF R (t)t = g( )t−1 .  (6.11)  With vr,s := vr,s ( ) = E[(T1+ )r (T2+ )s ] for non-negative integers r, s, g( ) =  (1 + ){3[ 21 π + arcsin( )] + (4 − ) 1 − v4,0 + 4v3,1 + 3v2,2 = 6(v2,0 + v1,1 ) 3{ 12 π + arcsin( ) + 1 − 2 }  g( ) is strictly increasing in  2}  .  ∈ [−1, 1]. From the equations in (6.11), we know  that for this truncated bivariate standard tν distribution, the convergence speed to the first order approximation is dominated by [VaRp (X1+ + X2+ )]−1 . When the correlation coefficient is larger — that is, when the dependence between margins is stronger — the second order approximation is more important. Some relevant observations for second order approximation of vector-valued CTE have also been reported in Example 3.2 of Joe and Li (2011). The effect from the tail heaviness of margins may also have subtle influence on the second order approximation, however we have to calculate higher moments in this case. A general conclusion about the effect of marginal heaviness on the second order approximation seems to be interesting and non-trivial to obtain. However, following the discussion in Remark 6.2, for the above bivariate t 158  distribution, the second order parameter ρ = −2. So the second order term will be negligible when [VaRp (X1+ + X2+ )] is large enough. In the following Example 6.5, we will illustrate by an example for a case of −1 < ρ < 0 that the second order term significantly improves the approximation and can not be ignored. Example 6.5 (Burr scale mixture of normal) Let (X1 , X2 ) := (Z1+ R, Z2+ R), where (Z1+ , Z2+ ) is left-truncated standard bivariate normal with correlation  =  0.9 and R ∼ Burr(a = 5, b = 0.4) (i.e., F R ∈ 2RV−2,−0.4 , refer to Example 6.2) with a scale parameter 1000. In Figure 6.2, the vertical axis is the ratio of CTE to VaR E[X1 |X1 + X2 > VaRp (X1 + X2 )] CTE := , VaR VaRp (X1 + X2 ) defined as a function of p. The true value is approximated based on a Monte Carlo simulation with a sample size of 107 , and the first and second order approximations are calculated based on Proposition 6.7. Figure 6.2: First/Second order approximations for Burr scale mixture of normal  2.5  CTE / VaR  1st order approximation true value 2rd order approximation 2.0  1.5  1.0 0.95  0.96  0.97  0.98  0.99  1.00  Probability level (p)  Example 6.6 Suppose T1 , T2 are independent and exponentially distributed with survival function FT1 (x) = FT2 (x) = 1 − e−x , Q = 1/R and independent of  159  T1 , T2 , F R (x) ∈ 2RV−α,ρ and (X1 , X2 ) := (RT1 , RT2 ). Then ∞  P[X1 > x1 , X2 > x2 ] =  e−(x1 +x2 )q FQ (dq) = ψQ (x1 + x2 ),  (6.12)  0  where FQ and ψQ are the distribution function and LT of Q, respectively. Take −1 −1 x1 = ψQ (u1 ) and x2 = ψQ (u2 ), and then −1 −1 −1 −1 P[X1 > ψQ (u1 ), X2 > ψQ (u2 )] = ψQ (ψQ (u1 ) + ψQ (u2 )) =: CψQ (u1 , u2 )  (6.13) Also, we have P[Xi > x] = ψQ (x) for i = 1, 2. Then ∞  E[X1 |X2 > t] = 0  ψQ (x + t) dx. ψQ (t)  (6.14)  For a random variable R following the inverse Gamma distribution with a shape parameter α and a scale parameter β, the density function fR (x) ∈ 2RV−α−1,−1 with an auxiliary function AfR (x) = βx−1 . We can derive from the density function and Proposition 6.5 that F R (x) ∈ 2RV−α,−1 and the auxiliary function AF R (x) =  αβ −1 α+1 x .  If we let Q ∼ Gamma(α, 1); that is, α is the shape parameter  and β = 1 is the rate parameter, since for the standard exponential random variable the a-th moment is Γ(a + 1), then we can get by Proposition 6.6 that E[X1 |X2 > t] ∼  1 1 t+ , α−1 α−1  t → ∞.  In another way, take ψQ (x) = (1 + x)−α , the LT of Gamma(α, 1) and from (6.14) it can be directly verified that E[X1 |X2 > t] = 1/(α − 1) + t/(α − 1), which is consistent with Proposition 6.6. This example just shows that Proposition 6.6 can also be used for a random vector where margins possess the tail behavior of ψQ (x) (i.e., P[Xi > x] = ψQ (x)) and the dependence structure is the Archimedean copula constructed by ψQ (x). This example is interesting since the tail behavior of the LT ψQ (x) affects both the tail dependence structure and the tails of margins. F R (x) ∈ RV−α implies that FQ (x) ∈ RVα (0). Then by the Karamata’s Tauberian Theorem (e.g., Theorem  160  1.7.1’ in Bingham et al. (1987)), ψQ (x) ∈ RV−α . That is, the margins are all regularly varying with index −α. Moreover, there is upper tail dependence for (X1 , X2 ), since ψQ (t) ∈ RV−α , the corresponding copula CψQ has lower tail dependence (refer to subsection 3.1 of Charpentier and Segers (2009)) and the copula for (X1 , X2 ) is the survival copula of CψQ . Based on (6.14), it seems that the second order property of ψQ (x) could also be related to the second order parameter of F R . We conjecture that under some conditions, if F R (t) ∈ 2RV−α,ρ and Q := 1/R, then ψQ (t) ∈ 2RV−α,ρ .  6.5  Concluding remarks  Second order regular variation provides a nice theoretical platform for studying second order approximations of limiting properties, like the asymptotic relationship between CTE and VaR that we have studied in this chapter. More importantly, many parametric distributions satisfy those theoretical assumptions so that the implementation of the main results is feasible. From the viewpoint of risk management, the study on risks at the sub-extremal level is more realistic and important, and we believe that the study involving the second order condition on asymptotic analysis for risk measures should be promising for this purpose. For future research, it will be interesting to study how marginal tail heaviness and tail dependence structures affect the second order approximations of risk measures. Also, the second order regular variation we have studied on risk measures — no matter for the single risk or for the multiple risks — is univariate second order regular variation. To get a more general result for multiple risks, we need to study the performance of risk measures under multivariate second order regular variation. Moreover, whether the second order parameter ρ is greater than −1 is critical to decide on the necessity of a second order approximation. So it will be interesting to develop a methodology to test H0 : ρ ≤ −1 verses H1 : ρ > −1.  6.6  Proofs  Proof of Lemma 6.2: Since F ∈ RV−α , we can write F (t) = t−α (t) where (t) is slowly varying and non-negative. By Definition 6.1, there is an auxiliary function  161  A and a constant c = 0 such that x−α (tx)/ (t) − x−α xρ − 1 = cx−α . t→∞ A(t) ρ lim  Therefore, lim  t→∞  (tx)/ (t) − 1 (tx) − (t) xρ − 1 = lim =c . t→∞ A(t) (t) A(t) ρ  Since the sign of A(t) can be adjusted by changing the sign of c due to Definition 6.1, if c and A(t) have the same sign, then we assume that c > 0 and A(t) > 0, and thus A(t) (t) > 0. By Theorem B.2.2 of de Haan and Ferreira (2006), then limt→∞ (t) = k with 0 < k < +∞ and k − (t) ∈ RVρ . So we can rewrite F (t) = kt−α (t) with limt→∞ (t) = 1 and also 1 − (t) ∈ RVρ . If c and A(t) have different signs, then let A(t) > 0, −c > 0 and then consider the function − (t) instead of (t). Also by Theorem B.2.2 of de Haan and Ferreira (2006), limt→∞ − (t) = −k with 0 < k < ∞. So F (t) = kt−α (t) with limt→∞ (t) = 1 and also (t) − 1 ∈ RVρ . Proof of Proposition 6.3: From Lemma 6.2, we can write F (t) = kt−α (t) with 0 < k < +∞ and limt→∞ (t) = 1. Then by Definition 6.1, with an auxiliary function A(t) and a constant c = 0, x−α (tx) − x−α (t) xρ − 1 F (tx)/F (t) − x−α = lim = x−α . t→∞ t→∞ cA(t) cA(t) (t) ρ lim  That is, lim  t→∞  (tx) − (t) xρ − 1 = . cA(t) (t) ρ  If c and A(t) have the same sign, then cA(t) (t) > 0 and by Theorem B.2.18 of de Haan and Ferreira (2006), we have for any , δ > 0, there exists a t0 = t0 ( , δ) such that for all t ∧ tx ≥ t0 , (tx) − (t) xρ − 1 − ≤ xρ max(xδ , x−δ ). −ρ[1 − (t)] ρ 162  (6.15)  A similar argument can be used for the case where c and A(t) have different signs, and inequality (6.15) still holds. Multiplying both sides of (6.15) by x−α finishes the proof. Proof of Proposition 6.4: Denote that t := VaRp (X), and the asymptotic relationship of (6.6) without the term η(t) can be easily derived by applying the Karamata’s theorem. Write F (t) := kt−α (t) and choose a(t) := −ρ[1 − (t)]/ (t). Then CTEp (X) ∞ t P[X  > x]dx =t 1+ P[X > t]  =t+  ∞  =t 1+ 1  =  ∞ t P[X  > x]dx tP[X > t]   P[X > ts] ds P[X > t]  ∞  = t 1 + a(t) 1  P[X > x] dx tP[X > t] t  P[X>ts] −α 1  P[X>t] − s ds + a(t) α−1  ∞ P[X>ts] P[X>t]  ta(t) α t+ + ta(t) α−1 (α − 1 − ρ)(α