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The index of dispersion Avelino, Edgar G. 1984

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THE INDEX OF DISPERSION By EDGAR G. AVELINO B.Sc, The University of British Columbia, 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FQR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES , Statistics Department The University of British Columbia We accept this thesis as conforming to the_ requifibd/t stan'cQrd' THE UNIVERSITY OF BRITISH COLUMBIA November 1934 © Edgar G. Avelino, 1984 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of GTA7~/S7?&'  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 'OCT- A3, /<?8y DE-6 (3/81) i i ABSTRACT The index of dispersion is a statistic commonly used to detect departures from randomness of count data. Under the hypothesis of randomness, the true distribution of this statistic is unknown. The accu racy of large sample approximations is assessed by a Monte Carlo simulation. Further approximations by Pearson curves and infinite series expansions are in vestigated. Finally, the powers of the individual tests based on the likelihood ratio, the index of dis persion and Pearson's goodness-of-fit statistic are compared. i i i TABLE OF CONTENTS PAGE ABSTRACT ii ACKNOWLEDGEMENT ix 1. INTRODUCTION 1.1 History of the Index of Dispersion 1 1.2 Purpose of the Paper 4 2. LARGE SAMPLE APPROXIMATIONS 2.1 Joint Distribution of X & S2 6 2.2 The Asymptotic Distribution of I 7 2.3 Description of the Monte Carlo Simulation 9 2.4 The x2 Approximation 14 2.5 Discussion 15 3. PEARSON CURVES 3.1 The Theory of Pearson Curves 19 3.2 Two Examples 21 3.3 The First Four Moments of I 24 3.4 Discussion 29 iv PAGE 4. THE GRAM-CHARLIER SERIES OF TYPE A 4.1 The Theory of Gram-Charlier Expansions 37 4.2 Discussion 42 5. THE LIKELIHOOD RATIO AND GOODNESS-OF-FIT TESTS 5.1 The Likelihood Ratio Test 49 5.2 Pearson's Goodness-of-Fit Test 54 5.3 Power Computations 55 5.4 The Likelihood Ratio Test Revisited 65 6. GONGLUSIONS 74 REFERENCES .....76 APPENDIX Al.l The Conditional Distribution of a Poisson Sample Given the Total 79 A1.2 The First Four Moments of I 80 A1.3 The Type VI Pearson Curve 7 A1.4 A Limiting Case of the Negative Binomial 88 A2.1 Histograms of I 90 A2.2 Empirical Critical Values 106 A2.3 Pearson Curve Critical Values 107 A2.4 Gram-Charlier Critical Values 109 V TABLE OF CONTENTS LIST OF TABLES PAGE TABLE 1: Expected Number of I*'s 10 TABLE 2 (A-D): Normal Approximation 1 TABLE 3 (A-D): x2 Approximation 16 TABLE 4 (A-D): Pearson Curve Fit with Exact Moments 31 TABLE 5 (A-D): Pearson Curve Fit with Asymptotic Moments 33 TABLE 6 (A-D): Gram-Charlier Three-Moment Fit (Exact) 43 TABLE 7 (A-D): Gram-Charlier Four-Moment Fit (Exact) 45 TABLE 8: Asymptotic Power of the Index of Dispersion Test -. 57 TABLE 9-10 (A-D): Power of Tests Based on A,I and X2 60 TABLE 11 (A-D): Power of Tests Based on A and I (n=50).. 66 TABLE 12: Number of Times (n-l)S2 ^ nX 68 TABLE 13 (A-B): Power of Tests Based on A,I and X2 (n=10) 70 TABLE 14 (A-B): Power of Tests Based on A,I and X2 (n=20) 71 vi PAGE TABLE 15 (A-B): Power of Tests Based on A and I (n=50).. 72 TABLE Al: The Ratios of the Moments of I and x2n_! 86 TABLE A2: Emperical Critical Values of I (Based on 15,000 Samples) 106 TABLE A3: Pearson Curve Critical Values (Exact) 107 TABLE A4: Pearson Curve Critical Values (Asymptotic) 108 TABLE A5: Gram-Charlier Critical Values (Three Exact Moments) 109 TABLE A6: Gram-Charlier Critical Values (Three Asymptotic Moments) 110 TABLE A7: Gram-Charlier Critical Values (Four Exact Moments) Ill TABLE A8: Gram-Charlier Critical Values (Four Asymptotic Moments) 112 vii TABLE OF CONTENTS LIST OF FIGURES PAGE FIG. A.l: Histogram of I (1000 Samples, n = 10, X = 3) 90 FIG. A.2: Histogram of I (1000 Samples, n = 10, X = 5) 91 FIG. A.3: Histogram of I (1000 Samples, n = 20, X = 3) 92 FIG. A.4: Histogram of I (1000 Samples, n = 20, X = 5) 93 FIG. A.5: Histogram of I (1000 Samples, n = 50, X = 3) 94 FIG. A.6: Histogram of I (1000 Samples, n = 50, X = 5) 95 FIG. A.7: Histogram of I (1000 Samples, n = 100, X = 3) 96 FIG. A.8: Histogram of I (1000 Samples, n = 100, X = 5) 97 FIG. A.9: Normal Probability Plot for I (1000 Samples, n = 10, X = 3) 9S v i i i PAGE FIG. A.10: Normal Probability Plot for I (1000 Samples, n = 10, x = 5) 9S FIG. A.11: Normal Probability Plot for I (1000 Samples, n = 20, x = 3) 100 FIG. A.12: Normal Probability 'Plot for I (1000 Samples, n = 20, x = 5) 101 FIG. A.13: Normal Probability Plot for I (1000 Samples, n = 50, X = 3) 102 FIG. A.14: Normal Probability Plot for I (1000 Samples, n = 50, X = 5) 103 FIG. Ail5: Normal Probability Plot for I (1000 Samples, n =.100, X = 3) 104 FIG. A.16: Normal Probability Plot for I (1000 Samples, n = 100, X = 5) 105 ix ACKNOWLEDGEMENT I wish to express my sincere appreciation to Prof. A. John Petkau who devoted precious time to the supervision of this thesis. Edgar G. Avelino 1 1. INTRODUCTION 1.1 HISTORY OF THE INDEX OF DISPERSION The index of dispersion is a test statistic often used to detect spatial pattern, a term ecologists use to describe non-randomness of plant populations. This is equivalent to testing that the growth of plants over an area is purely random, or equivalently that the number of plants in any given area has the Poisson distribution. Suppose then that we randomly partition some area by n disjoint equal-sized quadrats and make a count, x, of the number of plants in each quadrat. Under the hypothesis of randomness, Xj, ... , Xnwould have the Poisson distribution, For alternatives to complete randomness involving patches or clumping of plants, we would expect Var(X) > E(X), while for more regular spacing of plants, we would expect Var(X) < E(X) (see for example R.H. Green (1966)) . These properties lead quite naturally to considering the variance-to-mean ratio as a population index to measure spatial pattern. An estimator of the variance-to-mean ratio is the index of dispersion, defined as P(X=x) = e'V/xl , for x > 0 and x = 0, 1, 2, ... , for which E(X) = Var(X) = X. 1 , if I = 0 I = I s2/x if X > 0 n n where X = (1/n) F. X. and S2 = i=l 1 {l/(n-l)> z (X.-X)2, the i=l 1 2 unbiased estimators of E(X) and Var(X), respectively. (It is natural to define I to be 1 if X = 0 because under the null hypothesis, the variance-to-mean ratio equals 1.) Ever since G.E. Blackman (1935) used the Poisson model for counts of plants, the concept of randomness in a community of plants became a growing interest among ecologists. Although the index of dispersion was introduced by R.A. Fisher (Fisher, Thornton and MacKenzie, 1922), it was not until 1936 that it was first used by ecologists for the purpose of inference. A.R. Clapham (1936), using a x2 approximation for the distribution of the index of dispersion under the null hypothesis of randomness, found that among 44 plant species he studied, only four of these seemed to be distributed randomly, while over-dispersion (i.e. clumping) was clearly present for the remaining species. Student (1919) had already pointed out that the Poisson is not usually a good model for ecological data and in most cases, clumping occurs. This has been termed "contagious" by G. Polya (1930) and also by J. Neyman (1939). Ever since Clapham's paper, the use of the index of dispersion as a test of significance of departures from randomness has been extensive, not just for field data, but also in other areas (for example, blood counts, insect counts and larvae counts). Fisher et al (1922) showed that the distribution of the index of dispersion could be closely approximated by the x2 distribution with n-1 degrees of freedom. However, if the Poisson parameter, X» is small, or if the estimated expectation, X, is smal1, then the adequacy of the 3 x2 approximation becomes questionable. This is discussed by H.O. Lancaster (1952). Fisher (1950) and W.G. Cochran (1936) have pointed out that in this case, the test of randomness based on I should be done conditionally with given totals, £X.j. Since this sum is a sufficient statistic for the Poisson parameter X, conditioning on the total will yield a distribution independent of X. Hence, exact frequencies can be computed. The conditional moments of the index of dispersion are provided in Appendix A1.2. These moments are also given by J.B.S. Haldane (1937) (see also Haldane (1939)). Several people have examined the power of the test based on the index of dispersion. G.I. Bateman (1950) considered Neyman's contagious distribution as an alternative to the Poisson and found that this test exhibits reasonably high power for n>50 and mjm2^5, where mi and m2 are the parameters of Neyman's distribution. For 5^ns20, she found that the power is also high, provided that mim2 is large (in particular, mim2^20). Proceeding along the same lines as Bateman, N. Kathirgamatamby (1953) and J.H. Darwin (1957) compared the power of this test when the alternatives are Thomas' double Poisson, Neyman's contagious distribution type A and the negative binomial. They found that this test attained about the same power in each of the three alternatives. Finally, in a recent paper, J.N. Perry and R. Mead (1979) investigated the power of the index of dispersion test over a wide class of alternatives to complete randomness. They concluded that this test is very powerful particularly in detecting clumping, and they 4 strongly recommend the use of this test. Examination of the power of this test relative to other tests of the null may also be important. 1.2 PURPOSE OF THE PAPER The purpose of this paper is to examine the distribution of the index of disperion and compare its power to the power of other tests of randomness. We examine the properties of the index of dispersion and through these properties, attempt to answer such questions as: "How do we decide whether a given sample is significantly different from a Poisson sample?" and "How good is this test in detecting departures from randomness relative to other (perhaps reliable and well-studied) tests?" Answers to the first question could be based on constructing a rejection region R, where if I e R, we would tend to favor some other alternative. For example, if we wished to test the null hypothesis against alternatives involving clumping, then large values of I would provide evidence against the null hypothesis, and the rejection region would presumably be of the form I > C. For two sided alternatives, we would be interested in both large and small values of this statistic, say I < Cj or I > C2. We would also want to examine the chances of wrongly rejecting the null which in statistical terminology is called the probability of making a type I error or the significance level (or size) of the test. The constants C, Cj, and C2 are called critical values, and it is through these critical values that the rejection region will be constructed. 5 We then rephrase the question as: "Is there a method of determining the rejection region R at a given level of significance a?" As the true probability distribution of I is unknown, we first attempt to solve the problem through large sample approximations which lead to asymptotic critical values. We will show that the asymptotic null distribution of I is normal with mean 1 and variance 2/n. We can then use the critical values from the normal and determine how accurate these critical values are. This study is done through a Monte Carlo simulation. Similarly, the x2 approximation to the distribution of I is also examined. We also examine critical values obtained from approximating the null distribution of I by Pearson curves and Gram-Charlier expansions. To assess the "goodness" of the index of dispersion, we might be interested in determining how often we would correctly reject the null in repeated sampling. This is called the power of the test, the complement of this being the probability of making a type II error. With the negative binomial as an alternative to the Poisson, the power of I is then compared to the power of the Likelihood Ratio Test and Pearson's Goodness-of-fit test. 6 2. LARGE SAMPLE APPROXIMATIONS 2.1 THE JOINT DISTRIBUTION OF X AND S2 Suppose we choose a random sample of n disjoint equal-sized quadrats and make a count, X.., of the number of plants in the i^ quadrat. Let X^ X^ be independent identically distributed random variables with mean y and variance a2. Let = ECCX^-p)^] and suppose that vk<*. In particular, u\ - 0 and y2 = °2- As a consequence of the Central Limit Theorem, we have /n" (x-y) NCo,M2). (2.1) /?, (S2-y2) N(0,yit-y22). (2.2These results can be found in Cramer (1946, pp. 345 - 348). Similarly, the Multivariate Central Limit Theorem implies that /n(X-p) and /n(.S2-y2) converge jointly to a bivariate normal distribution with mean vector JD and variance-covariance matrix (l/n)E, where 11m nC0V(X,S2) lim nC0V(X,S2) n -*• « yit - V24 l .e. •^n(X-y) ^(S2-y2) N(0,E) (2.3) Assuming that y = 0, we have C0V(X,S2) = E(X-S2) = (n/(n-l)){E(XU2) - E(X3)} where U2 = (l/n)EX? . Hence E(XU2) = (l/n2)E{EX.3 + EE X.X.2} 1 ifj 1 J = y3/n , since by independence, the double sum has zero expectation. Similarly, E(X3) = y3/n2 from which it follows that C0V(X,S2) = y3/n + 0(l/n2). \2A) From (2.3) and (2.4) we have for large n that N -^2 , (1/n) V2 ^3 V3 Vii-V2' 2.2 THE ASYMPTOTIC DISTRIBUTION OF I We compute the asymptotic distribution of S2/X using the "delta method". It will be seen that the asymptotic distribution of I is the same as that of S2/X\ Let g(x,y) = y/x, so that S2/X = g(X,S2). Assuming that 3g/8x and 8g/3y exist near the point (y,cr2), (note that this requires the assumption that y > 0), we can expand g(X,S2) in a Taylor series about (y,o2) and have g(X,S2) = g(y,o2)+(X-y)gx(y,o2)+(S2-a2)gy(y,o2)+ ... Let U(n)'=(5(,S2) and b = (y.a2)'. Then (i) u(n) -P—b ; (ii) /n"(U(n)-b) N(0,E). 8 The result of the delta method (see, for example, T.W. Anderson (1958, pp. 75 - 77).). is that /n"((S2/X) - (a2/p)) -1 N(0 , where <|>b' = (8g/8x,3g/8y) evaluated at (p,a2). Under the stated assumptions, we have for large n, S2/X * N(a2/p , (l/nj^'z^). (2.5) After some matrix calculations, we get • b'z+b = vi3/^ - 2y2y3/y3 + hk-V22)/v2~. (2.6) So far, all of the results hold regardless of the underlying distribution of the X's. If we now assume that Xj,...,X ~ P(x), then v = E(X) = X , p2 = Var(X) = X , P3 = X and Pit = 3x2 + X . Substituting these into (2.6), we have that • b'J:*b = (1A) - (2A) + C(3X2+X) - x2]/x2 = 2, and hence from (2.5) that S2/X » N(l , 2/n) . The asymptotic null distribution of I is easily seen to be the same as that of S2/X since P{|I-(S2/X)| > e} = P{X=0) = e"nX for any e>0. This probability approaches 0 as n —*• » , and hence I « N( 1 , 2/n) . 9 Note that the 0(.l/n) approximation to the variance of I is inde pendent of the parameter X. This would be useful in practice because the source of error in estimating X by the maximum likelihood estimator t would not have to be introduced. We should note however that the inclusion of higher order terms will introduce this dependence. 2.3 DESCRIPTION OF THE MONTE CARLO SIMULATION To answer the question of how well the asymptotic critical values work, we perfromed a Monte Carlo simulation when the underlying distri bution of the X's is Poisson. Fifteen thousand samples of n Poisson random variables were generated for n = 10,20,50,100 and for X = 1,3,5,8 and fifteen thousand indices of dispersion were computed for each pair n and X. The 1%,2.5%,5% and 10%. quantiles for each pair n and X are given in Table A2 . With such a large number of samples, these critical values may be regarded as exact and they assist in assessing the accuracy of the asymptotic critical values. Given a nominal significance level a, two-sided rejection regions were constructed with a/2 in each tail. Using the asymptotic normal critical values, the rejection regions used were the following: R.01 = {I*: |I*| > 2.58} R.05 = {I*: |I*| > 1.96} R.10 = {I*: > 1.64} R.20 = (I*: |I*| > 1.28} where I* = (I-1)//(2/n) and where R^ denotes the rejection region at the nominal significance level a. To test the accuracy of the normal critical values, we merely count the number of I*'s that fall in R This a 10 would then give us an estimate p, of the true significance level p. Now, p = C# of I*'s e R )/15,0Q0. Since the number of I*'s e R a a is binomi'a-1 ly distributed (with parameters N=15,000 and p), the A standard error of p is SECp) = / p(.l-p)/15,000 . We then might conclude that the distribution of I is well approximated by the normal if p is within one standard error of the nominal signifi cance 1evel a. To assist the reader in interpreting the results, we supply a list of how many I*'s would be expected in each tail of the rejection region if the true significance level corresponding to each tail was identically equal to a/2, one-half the nominal significance level. Table 1: Expected Number of I*'s a ' {a/2 ± SE(a/2)} • 15,000 0.01 75 ± 9 0.05 375 ± 19 0.10 750 ± 27 0.20 1500 ± 37 The results, summarized in Tables 2(A-D), are shown in the following pages. The entries in the "I<L" and "I>U" columns are the number of I*'s that lie to the left and right of the lower and upper normal cri tical values, respectively. We immediately notice that the normal approximation is very poor even for n as large as 50. The lower critical values are much 11 NORMAL APPROXIMATION Table 2A (a = O.Ol) X 1 3 5 8 I <L I>U j KL I>U : KL I>U I <L I>U I 10 0 342 1 0 293 0 . 286 0 276 I 20 0 280 0 243 0 227 0 213 § 50 9 220 9 197 13 175 12 182 100 19 184 27 145 22 134 26 Table 2B (a = 0.05) X 1 3 5 8 n KL 1>U : KL I>U KL I>U 1 KL i>u 1 10 8 713 3 645 o 670 2 646 1 20 50 678 74 633 . 76 607 72 587 1 50 148 587 1 183 546, ] 199 533 197 515 I 100 I 199 538 1 248 506 244 490 1 2" 481 I 12 Table 2C ( a = 0.10) ] L 3 8 n 1 [-• KL I>U f; KL I>U ' Kb I>U KL I>U i 10 114 951 ' 142 974 145 1011 149 1007 | 20 ' 283 1010 368 964 345 983 359 986 I 50 457 962 519 909 527 938 541 903 100 554 943 570 872 • 570 -859 592 862 Table 2D (a = 0.20) n I KL I>U I KL I>U I KL I>U KL I>U 10 ' 596 1520 I 962 1571 • 953 1633 958 1607 20 1008 1447 |a 180 1627 11212 1643 1221 1648 50 1216 1612 1.1338 1589 |,1334 1587 1324 1584 100 1386 1528 I 1377 1522 I 1322 1576 1355 1564 13 too liberal. For the cases ns5Q, the total number of I*'s in the re jection region is close to the total number we would expect to be rejected, hut the significance level in each tail is nowhere near a/2. The probability of falsely rejcting the null would be too low in the lower tail and too high in the upper tail. There is obviously a prob lem of skewness in the distribution. Too many observations lie in the right tail implying that the distribution of I is positively skewed. Notice that for fixed x and increasing n, the number of I*'s rejected in each tail becomes more equal. However even for n=100, the lower critical values are still conservative in all significance levels while the upper critical values are too liberal. For fixed n and in creasing X on the other hand, no such pattern is obvious. Thus, it appears that the normal approximation is really only satisfactory for n>100 and this will not suffice for practical work. An examination of the probability plots and histograms (see Appendix 2.1) provides more detail. One might hope to find an improvement to this approximation and one approach taken to improve the approximation is through infinite series expansions (e.g. Edgeworth, Gram-Charlier, Fisher-Cornish expansions). The trade-off for having such an improvement is the requirement of higher order moments; and these higher order moments will surely have a dependence on X. More of this will be seen in later chapters. For the moment, we abandon the normal approximation and move on to another simple large sample approximation. 14 2.4 THE x2 APPROXIMATION As seen in section 2.2, the probability under the null that I and S2/X differ by an amount bigger than E(E>0) is e nX which approaches 0 as n —• ». It is therefore sufficient to consider an approximation to the distribution of S2/X. At a first glance, we might suspect that S2/X has some relation ship with the x2 distribution, for it is well known that if Xj, • is a random sample from the normal distribution with mean y and variance a2 , then (n-l)S2/a2 ~ x2 . n-1 In our case, the X's are Poisson and Var(X) is only approximated-by 52 = X. However, it would not be surprising that the null distribu tion of (n-l)S2/X could be well approximated byx2n ^ for large n. A clearer motivation of this is outlined below. Consider the following one-way contingency table: X2 • • • Xn X. E.: X J X — X nX* The entries in the cells of the first row are just the observed counts themselves, having row total X., and the entries in the second row are the estimated expected counts, X. (Note that this contingency table differs from the ordinary contingency table where observations are free to fall in any one cell. In our contingency table, we have 15 one cell for each count. However, if we considered only those sampling experiments that produced the same order of experimental results in add ition to the same marginal totals, the methods of the ordinary contingen cy tahles still apply.) The goodness-of-fit statistic is formed by summing up over the columns, the square of the difference between the observed and the expected values and dividing this by the expected value. This gives us " CX.-X)2/X" , i=l 1 • which is precisely (n-l)S2/X. Providing the E.'s are not too small J (for example, E.>5 for all j), the distribution of the goodness-of-fit J statistic might be expected to be well approximated by x2p ^ for large n. This motivation is due to P.G. Hoel (1943). In his paper, he approximated the moments of S2/X under the null hypothesis by power series expansions, correct to 0(l/n3), and showed that the first four moments of (n-li)S2/X were in close agreement with those of the x2n j distribution. 2.5 DISCUSSION Returning now to the simulation study, we recall that since the normal distribution is symmetric, it could not account for the skew-ness of the distribution of I. On the other hand, since the x2n_| distribution is skewed, one might expect it to perform better than the normal approximation. So as not to obscure the comparison of the two approximations, the same 15,000 samples generated for each case were used. The results are displayed in Tables 3(A-D). 16 X2 APPROXIMATION Table 3A ( a = 0.01) X 1 2 1 5 8 n I ' KL I>U j I>U I KL I>U J KL I>U 10 j 44 72 1 59 81 77 . 83 77 60 20 39 100 69 93 63 76 | 66 78 1 50 • 50 93 69 93 73 9b 75 : 82 100 50 102 72 72 72 • 67 77 69 . Table 3B ] (a = 0.05) L. 3 X 5 8 n • .' KL I>U KL IMJ KL I>U KL I>U 1 10 182 342 . 356 354 379 356 385 360 20 213 408 336 385 1 340 365 359 360 50 : 285 408 350 405, ." 363 380 - 373 365 1 100 ; 309 419 349 396 : 359 373 |: 370 367 9 17 Table 3C (ct = 0.10) KL 10 B 587 I>U KL 713 i 633 I>U KL 707 730 I>U 1 KL 733 1 777 I>U 720 20 1 537 741 9 725 50 I 607 100 I 676 774 807 fl 729 775 796 B 693 757 704 791 I 750 708 684 752 I 765 738 R 709 756 756 733 Table 3D (a = 0.20) 5 n KL I>U KL IMJ | KL IMJ IMJ 10 968 1083 1383 1429 1 1464 1478 : 1489 1469 20 - 1225 1350 : 1478 1504 1- 1468 1507 1517 1539 I 50 1463 1534 1448 1528 : 1457 1505 1468 1495 1 100 1 1440 1489 t 1432 1472 1411 1508 I 1454 1502 1 18 The x2 approximation clearly gives a better fit to the null distribution of I than the normal. Most of the entries in the cells fall within the range of values that one would expect to see. Notice that these tables display a similar pattern, namely that symmetry be tween the "I<L" and "I>U" columns becomes more apparent with increasing n and fixed X (and with increasing X and fixed n). This pattern appears with increasing a too. However, there seems to be more room for improve ment for the cases n^20 and x<3. In fact, even for n=100 and x=l, the lower critical values tend to be too conservative while the upper criti cal values tend to lead to rejection too often. (In the ecological con text however, this would cause no serious problems. One can simply take larger quadrats to ensure that the mean number of plants in each quadrat is larger than 1.) For the cases n^20 and X^3, the x2 approximation i gives a very reasonable approximation to the null distribution of I, and leads to a pleasantly simple method of constructing rejection regions. ' As stated in the previous section, one way of improving these large sample approximations is through an infinite series expansion of the true density of I. Another technique commonly used is approximations by Pearson curves, which will require the first four moments of I. 19 3." PEARSON CURVES 3.1 THE THEORY OF PEARSON CURVES The family of distributions that satisfies the differential equation d(log f)/dx = (x-a)/(b0+b1+b2x2) C3.1) are known as Pearson Curves. Under regularity conditions, the constants a.bo.bj and b2 can be expressed in terms of the first four moments of the distribution f; see Kendall and Stuart(1958, vol. 1, p. 149). Karl Pearson (.1901) identified 12 types of distributions each of which is completely determined by the first four moments of f. It is convenient to rewrite-the denominator as B0 + Bi(x-a) + B2(x-a)2 for suitably chosen constants BQ.BJ and B2, and hence (3.1) may be written as d(log f)/dx = (x-a)/{B0 + B^x-a) + B2(x-a)2 . (3.2) As in (.3.1), the constants in C3.2) are functions of the first four i moments of f(.x). By integrating the right-hand side of (3.2), an ex plicit expression can be obtained for f(x). The criterion for determining which type of Pearson curve results is obtained from the discriminant of the denominator in (3.2). This criterion is given by: K = Bi/4B0B2. (.3.3) 20 Defining 5i = V3/V2 and B2 = VH/V2> where the y.'s are the central moments for f(x), the constants Bo.Bi and B2 can be ex pressed in terms of 3i and B2. The criterion K then becomes K = 61(32+3)2/{4(2e2-3Bi-6)(432-3Bi)}. (3.4) For example, a value of K<0 gives Pearson's type I curve, also called the Beta distribution of the first kind. In this case, f(x) = kx^U-x)^1 , for 0 < x s 1, where the constants k,p and q are functions of the first four mo ments. If 1<K<«, then we get Pearson's type VI curve, also known as the Beta distribution of the second kind. Here, Hx) = kxp-1/U+x)p+q , for 0<x<°°. The following is a summary of the steps one would take when approximating by Pearson curves. ' Let g(X_) be a statistic whose null distribution we wish to approximate by Pearson curves. The first step is to compute the first four moments of g(X) which will depend on the para meters of the null distribution of the X's (if the parameters are not specified, they may be estimated by the maximum likelihood). Then &i and 62 • can be computed from the moments. From here, either one of two routes can be taken. If critical values are all that are required, then the Biometrika tables pub-21 1ished by Pearson and Hartley (1966) can be used. The critical val ues are tabulated for a wide range of values of /Bi and 82, and if necessary, linear interpolation along rows and columns is sufficient. We should note that when using critical values from the Biometrika tables, one should keep in mind that those critical values are all standardized. So if X denotes the ot level critical value from the ct tables, then the. appropriate a level critical value to use for the test is x = (/M!)X + ji. a a H The other alternative is to compute K from (3.4) and determine the type of Pearson curve to be used. If the resulting distribution is not too uncommon, the parameters of the distribution can be com puted. The text by W.P. Elderton and N.L. Johnson (1964, pp.35-46) gives an excellent treatment of this situation. Once the Pearson curve is completely determined, critical values can usually be ob tained from the computer. In particular, the IMSL library provides critical values for a wide class of distributions. 3.2 TWO EXAMPLES Before applying Pearson curves as an approximation to the null distribution of I, we discuss briefly two of the examples from the paper by H. Solomon and M. Stephens (1978), where the accuracy of critical values obtained from a Pearson.curve fit is examined. 22 n 2  Example 1: Let Q (c,a) = E c.(X. + a.) , n i=l 1 1 1 where c = (c 1' • • • 9 c )' and a_ = (aj,...,an)' are vectors with constant components, and the X.'s come from a standard normal distribution. The exact moments of this statistic are known to any order. In fact, the rth cumulant < . is There is a long literature on obtaining the critical values for dif ferent combinations of n,c_ and a/. These critical values are tabula ted in Grad and Solomon (1955) and in Solomon (1960). Much mathema tical analysis was used to obtain.these critical values and an exten sive amount of numerical computations were made so accurately that all the critical values can be regarded as exact. Pearson curve fits were obtained for different values of the constants and critical values were obtained by-quadratic interpola tion from Biometrika tables. The results were that the Pearson curve critical values agreed very closely with the exact critical values for the upper tail, but there was no close agreement at all between the two critical values in the lower tail of the distribu tion. Now, Pearson curves can also be obtained when the first three moments and a left endpoint of the distribution are known. (See for r n r 2 (r-1): E c.(l+ra.). i = l 1 1 23 example, R.H. Muller and H. Vahl (1976) and A.B. Hoadley (1968) ). Solomon and Stephens proceeded to do this three-moment fit and found that the fit in the lower tail was improved considerably, but this approach made the fit in the upper tail less accurate. They point out however that whenever four moments are available, then these four mo ments should be used for the fit as it is the upper tail of the dis tribution that is usually of more importance in practice. Of course, in our case, depending on whether we are concerned with clumping or regular spacing of plants, either tail of the distribution might be of interest. Example 2: Solomon and Stephens considered the statistic U = R/S, where R is the range and S the standard deviation of a sample from the standard normal distribution.' For n=3, the density of. U is known: . f(u) = (3/TT){1-(U2/4)}~(1/2\ for /3~< u < 2. The Pearson curve turned out to be a Beta distribution of the first kind and had the form g(u) = 0.9573/{(u-1.7324)°-0101(2.000-u)0-4970}, for 1.7324 £ u < 2.000. First we notice that while the true distribution of U is bell-shaped, the Pearson curve is U-shaped. However, notice that the Pearson curve fit gave the correct left and right endpoints of the distribution, at least to three decimal places. Finally, Solomon and Stephens found that the Pearson curye crticial values agreed very well with the exact critical values in both the lower and upper tails of the distribution. Given that the Pearson curve is U-shaped, the accu rate fit obtained in both tails of the distribution is extremely sur-24 prising' These two examples illustrate the usefulness of Pearson curves as a means of approximating perhaps not so much the distribution, but the critical values. 3.3 THE FIRST FOUR MOMENTS OF I Computing the first four moments of I is no easy task since I involves a random variable in its denominator, namely X. However, we can express the expectation of I , for k = 1,2,3,4, as the expec-n tation of the conditional expectation given the total X. = E X. (or given X). Now jk J \ ^ x = o (S2/X)k if X > 0 If follows that E(Ik|X.) = \ 1 if X. = 0 E{(S2/X)k|-X.} if X. > 0 UX.=0} + E{(S2/X)k|X.} i {X.>0}, where i{A} is an indicator function equalling 1 if A is true and 0 otherwise. Hence V - Ed") = E{E(Ik|X.)} = P(X.=0) +l=l E{(S2/X)k| X.=j}-P(X.=j) = P(X.=0) + E+{Er(S2/X)k!xJ> (3.5) where E+ denotes an expectation over the marginal distribution of X. restricted to the positive values of X. . Thus we may write the kth_ 25 moment of I as uk' = PtX.=0) +E+(Cl/X)k.EC(S2)k | X]}. (3.6) Hopefully, the conditional expectation,.which will depend on X, -k might cancel ,off the X in the denominator, and hence computation of the unconditional expectation will be relatively easy. Another difficulty arises in computing the conditional expec tation itself, which involves expanding k n k (S2)K = {[l/Cn-l)] E (X.-X)2}\ for k = 1,2,3,4. i=l 1 The. conditional expectation of this random variable will involve moments and product moments of (X^.X^,...,Xn)|X. up to the eighth order, and hence if we choose to compute moments from the moment generating funtion, we would require mixed partial derivatives of the moment generating function of (X^.Xg,....^ )|X. up to the eighth order. Fortunately, when the underlying distribution of the X.. 's is Poisson, the distribution of (X^.Xg xn^X* has a ^ariV'v simple form, reducing to a multinomial distribution with parameters X. and p.. = 1/n, for i=l,2 n; i.e. (X1,X2 xn)lx- ~ Mult (X.,1/n,...,1/n). This result, the derivation of which is provided in Appendix Al.l, facilitates the derivation of the conditional moments E{(S2/X) |X.=x.} for x.>0. These are provided in equations (Al.3)-(Al.6) of Appendix 26 A1.2 and lead via (3.5) to the following, expressions for the first four raw moments of I, the index of dispersion; P(X.=0) + P(X.>0) = 1 P(X.=0) + {(n+l)/(n-l)}P(X.>0) - {2/n(n-l)}E+(l/X) P(X.=0) + {(n+l)(n+3)/(n-l)2}P(X.>0) +{l/(n-l)2}-{(-4/n)[l-(6/n)]E+(l/X2) - 2[l+(13/n)]E+(l/X)} P(X.=0) + {(n+l)(n+3)(n+5)/(n-l)3}P(X.>0) + {l/(.n-l)3}{4nCl+(5/n)][l-(17/n)]E+(l/X) - (4/n2)(2n2+53n-261)E+(l/X2) -- (8/n3)(n2-30n+90)E+(l/X3)} . (3.7) From these expressions, we see that we have not overcome the problem of evaluating the expectation E (1/X ) for k = 1,2,3. We have taken two approaches in evaluating these expectations. The practical approach is to express these expectations as integrals, and evaluate these integrals by asymptotic expansions. This is done in Appendix A1.2; the resulting expressions for the raw moments, correct to 0(l/n'»), are provided in equation (A1.7). The central moments, correct to i ^3 i Pi* 27 OU/n1*), are then immediate: u = 1 (exact), u2 ~ 2/n + C2/n2){l-(l/x)} + (2/n3){l-(l/x)-(l/x2)} + (2/n£M{l-a/A)-(l/X2)-(2/X3)} + 0(l/n5), y3 ~ (l/n2){3+(4/x)} + Cl/n3){16-(24/x)} + (l/n\K24-(52/x)-(3/x2)-(4/X3 )} + 0(l/n5), MK ~ 12/n2 + (l/n3){72+(72/x)+(8/x2)> + ,(l/nlt){180-(240/x)-(228/x2)+(15/x3)} + 0(l/n5). (3.8) Using the definition of-Si andB2 , we can also express these in a similar expansion: ~ (2/n){2+UA)}2 + (2/n2){4-(16/X)-C3/X2)+C3/X3) + (2/n3){4-(ie/x)-(7/x2)-(17/X3)+(7/xl*)+ ...}. (3.9) B2 ~ 3 + (2/n){6+(12/x) + U/X2)} + (2/n2){6-(.36/x)-(5/X2)+(4/X3)) + (3.10) 28 The accuracy of these approximations can be assessed by computing the moments "exactly". By this, we mean computing the moments to a reasonable degree of accuracy. To do this, we can approximate the A. L infinite series in the expressions for the exact moments by N ' partial sums, S^, where N, the number of terms in the partial sum, is chosen so that the difference between the true and approximated values is no bigger than 10~6, say. A geometric bound on the error is shown below. oo Let S = (l/n)E+(l/X) - E (l/k)e~9ek/k'. k = 1 Where 6 = nX. We want to determine N such that S-S.. 5 10~6. Now, N OO S-S.. = z (l/k)e"eek/k'. ,N k = N+l co <; E e"0ek/k'. . k <= N+l = {e"e8N+1/(M+l)'.}{l+Ce/(N+2)] +[e2/(N+2)(N+3)] + Ce3/(N+2)(N+3)(N+4)] + ...} ^ {e'0eN+1/(N+l):}{l+(e/N)+(e/N)2+(0/N)3+ ...} = {e"eeN+1/(N+l):}{i/[l-(e/N)]}, if e<N. 29 We therefore want to choose N so that (i) N > nx and (Ii) S-SN <; IO"6 We note that this same value of N can be used for E+(l/X2) and E+(l/X3) since convergence is faster in these cases. The importance of a good asymptotic expansion is clear; one would not want to compute partial sums when fitting by Pearson curves. While computation of "exact" moments may be relatively inexpensive for small values of e, it can get quite expensive for larger values of e, and furthermore, over-flow problems will occur in these cases. The accuracy of the asymptotic expansions is dis cussed in the next section. 3.4 . DISCUSSION Using the known values of X, we can compute exact and asymptotic moments and hence obtain two Pearson curve fits for the simulated data. The Pearson curves obtained are the following: i) Using asymptotic moments (up to the fourth order) a type IV fit was obtained for the case x=l (for all n) and a type VI for all other cases, ii) Using exact moments, the same types were obtained except for the case n=10andx=l where the fit turned out to be a type VI. The type IV Pearson curve is not a common distribution. f(.x) has the form f(x) = k{l+(x2/a2 )}"m exp{-b arctan (x/a)}. 30 Since the critical values from this distribution cannot be obtained from the IMSL library, all the type IV critical values were obtained from Biometrika tables (Pearson & Hartley, (1966)). However, the critical values from a type VI curve can be obtained from IMSL. The algorithm for determining the form of the density is outlined in Appendix A1.3. Using the same 15,000 samples for a given n and x , a Monte Carlo study was done to assess the Pearson curve fits. The reults of the study are presented in Tables 4(A-D) and Tables 5(A-D). As seen from the tables, the number of rejections from a Pearson curve fit are close indeed to the ideal number of rejections listed in Table 1. The cases of main concern (n^20 and x<3) seem to be satisfactory except for"the case n=10 and x=l where the critical values tend to reject too often. However, a definite improvement from the x2 approximation is clearly present for these cases. While the lower x2 critical values tend to be too conservative, the Pearson curve fit has corrected for this - however, it has over-corrected, as now, the lower critical values of the Pearson curves tend to be too liberal! This is apparent in all the significance levels considered. Note how similar the tables obtained using exact and asymptotic moments are. This indicates that the asymptotic expressions for the moments, when used up to the;fourth order, are fairly accurate. The excellent performance of the asymptotic moments is indeed encouraging for its use in applications. Even for n as low as 10, the asymptotic values of ^2^3 ancl m» were correct to the first 4,3 and 2 decimal places, respectively. 31 PEARSON CURVE FIT WITH EXACT MOMENTS Table 4A (a = 0.01) X 1 3 5 8 n KL I>U I I<L I>U KL I>U KL I>U 10 • 109 72 59 77 69 74 • 76 59 20 j 67 64 71 82 70 ' 75 69 76 50 : 87 63 85 84 «• 84 82 79 100 64 75 84 63 79 65 79 67 Table 4B ] (a = 0.05) L 3 X 5 8 n KL I>U 1 KL I>U KL I>U KL I>U I 10 587 410 : 384 399 384 376 391 365 1 20 ' 401 407 : 384 372 364 362 1 381 360 I 50 ; 401 371 383 397 , | 383 371 384 352 1 100 \ 389 -393 366 380 379 368 - 380 I 364 1 32 Table 4C (a = 0.10) X 13 5 8 n KL I>U I KL IMJ : KL I>U IMJ 10 . 596 821 .. 758 747 I 780 . 760 790 744 20 t 682' 785 786 778 - 769 792 779 763 50 : 751 796 761 770 733 762 774 755 100 759 763 :. 728 749 703 735 724 730 Table 40 (a = 0.20) 1 3 X 5 8 n l KL IMJ : KL IMJ : KL IMJ KL IMJ I 10 , 1636 1514 • 1578 1510 j : 1549 1519 1519 1504 20 . 1424 1447 [1520 1534. [ : 1547 1537 1535 1543 50 • 1576 1561 1531 1538 . 1476 1511 1489 1505 100 .. 1502 • 1489 1470 1477 ' 1426 1511 j 1467 1502 33 PEARSON CURVE FIT WITH ASYMPTOTIC MOMENTS Table 5A (a = O.Ol) X 1 3 5 8 KL I>U KL I>U I KL I>U | KL I>U • 10 44 72 59 77 77 . 75 1 87 59 j 20 67 ' 61 I 71 82 70 75 • 69 76 50 84 64 . 85 84 79 84 ; 82 81 ' 100 71 74 -84 63 79 : 65 79 67 Table 5B (« = 0.05) X 1 3 5 8 n : KL I>U KL I>U I>U KL I>U I* 10 J 384 410 j 384 399 [ 397 374 405 369 20 ; 401 407 1 384 372 . '•: 364 361 381 360 50 f 401 381 I; 383 397, . 383 ; 371 ; 386 '352 100 : 394 384 1 366 384 383 368 :379 364 1 Table 5C (a = 0.10) X 1 3 5 8 n . ,' KL I>U r KL IMJ j: J<L I>U KL I>U 10 • 596 821 758 747 780. 756 806 744 I 20 I 683 785 786 778 763' 782 765 1 50 i* 744 796 . 761 770 I'" 733 762 778 755 1 100 [ 759 761 : 728 744 " 703 734 730 733 I Table 5D (a=0.20) 1 3 X 5 8 n KL I>U 1' KL I>U I>U KL I>U B 10 1636 2233 1578 1510 i 1547 1516 1519 1500 20 " 1424 1447 1; 1520 1534. ) 1547 1537 : 1533 1542 50 •1576 1573 . 1531 1538 j ; 1478 1511 ; 1489 1505 \ 100 1511 1489 1 1470 1468 . 1422 1507 Jf 1468 1501 35 The critical values obtained using exact and asymptotic moments may be found in Tables A3 and A4 respectively. With Pearson curve critical values now available, two issues come to mind: (i) While the approximate values obtained from Pearson curves . clearly improve upon those obtained with the x2 approximations for n<20 and/or X<3, is it worthwhile going through the Pearson curve algorithm, computing asymptotic moments, determining the Pearson curve and then obtaining the critical values, as opposed to simply going to x2 table and reading off the critical values? (ii) Are the Pearson curves still better when we replace X by the maximum likelihood estimator x=X? No attempt was made to examine the second question, although for large sample sizes, we would expect that Pearson curves would still be better. In answer to the first question, if accuracy of critical values is of primary importance, then we might favor Pearson curves. The asymptotic expressions for the moments are now known to be accurate, and once the moments are computed'" from these expressions (to the fourth order), $i and B2 are deter mined and the Biometrika Tables (Pearson and Hartley (1966)) provide us with the critical values. If on the other hand, the criterion K given in (.3.4) results in a not too uncommon distribution, then the critical values may be obtained from the computer. We reiterate that in this case, the explicit form of the density has to be derived. Alternatively, given the values of n and X, one can obtain critical values through interpolation from the tables 36 provided in the appendix. Although the accuracy of interpolating from these tables has not been assessed, the Pearson curve algorithm is smooth and presumably, a simple linear interpolation will suffice. 37 4. THE GRAM-CHARLIER SERIES OF TYPE A 4.1 THE THEORY OF GRAM-CHARLIER EXPANSIONS In mathematics, a typical procedure for studying the properties of a function is to express the function as an infinite series. Two types of series that immediately come to mind are Taylor series (or power series) and Fourier series. While these two series express a function as a sum of powers of a variable or as a sum of trigono metric functions, we will instead consider expanding the true density of I as a sum of derivatives of the standard normal density. One can then think of such an expansion as a correction to the normal approximation that was examined in Chapter 2. Let <f>(x) be the standard normal density. • Cx) (1/^) expC-x2/2), Then + '(x) -x<(. Cx) • "(x) (xMHCx) <J>(3)(x) -Cx3-3xH(x) *(lt)(x) 6x2+3)*(x) In general, »• • (4.1) 38 The polynomials H.(x) of degree j are called the Tchebycheff- Hermite polynomials. By convention, <l>^(x) = <f>(x), i.e. HQ = 1. Some important properties of these polynomials are that (i) HV(x) = jH.^Cx) '0 , k*j Cii) 2 Hj(.x)Hk(x)cbCx).dx =j k'. , k=j v. i.e. the Tchebycheff-Hermite poloynomials are orthogonal. Ciii) _/ Hj(xHCx)dx = -hVj^UMx). The proof of (i) involves expanding cp(x-t) = cbCx)exp{tx-Ct2/2)} in a Taylor series about t = 0. This yields the equation exp{tx-U2/2)} = ~ Ctj/j:)H.(x). Substituting in the series for the exponential term and redefining the index of the summation gives the desired result.; (ii) follows from (i) by substituting in the expression for H^(x) (k) from (4.1) in terms of <|> (x) and performing successive integration by parts, (iii) follows immediately from (4.1). Suppose then that a density function f(x) can be expanded in an infinite series of derivatives of • (x): f(x) =' i c.H.CxH(x). j=0 J J (4.2) 39 The conditions for this series expansion to be valid can be found in a theorem by Cramer (1926). The conditions are that: co (i) / (df/dx)exp(_-x2/2)dx converges, and (ii) f(x) —* 0 as x —>• ±°> . To find the coefficients c, multiply equation (4.2) by H.(x) J K and integrate from -°» to » and use the orthogonality property. CO CO CX> / f(x)H (x) dx = Y z c.H.Cx)H. tx)*Cx)dx K .» j=o JJ K OO OO = tloc.H.tx)Hk(x)*(x)dx (interchanging the sum and the integral is justified since p(.x)«<j>(xl is always bounded, for any polynomial pCx) of finite degree.) The c can then be expressed in terms of the moments about the origin. We list the first five coefficients below/. c0 = i Cj = u c2 = Ll/2)(ix2'-l) 40 c3 = (i/eKiia'-y) ch = (l/24)(yi+ ,-6u2,+3). For the purpose of computing critical values, it is convenient to express the cumulative distribution function (CDF) F(x), in a similar series. F(x) = /X f(x) dx —CO = /x{c{>(x) + ? c.H.(xHCx)} dx j=l J J = $(x) - Z c .H. ,(x)<f>(x}» where $ is the standard j=l J J_i normal CDF. This series is called the Gram-Charlier series of type A (Kendall and Stuart (1958), vol. 1, pp. 155-157). Let X = (I-l)/^. Then Ci= c2= 0, and the Gram-Charlier series of type A for the CDF of X is given by F(x) = $(x) - <Kx){c3H2(x) + c^Cx) + ...} where c3 = (l/6)p3* = Cl/6)E{(I-l)3/n23/2} = (l/6)y3/M2 3/2 = (l/6)/3T . Similarly, cu = (l/24)(.B2-3). Now consider using partial sums of the Gram-Charlier series as 41 an approximation to FCx). (Note that the one-term approximation is merely the normal approximation.) In particular, suppose we use the first two terms of the series to approximate F(x). FCx) * *(x) - Cl/6)(/Bl)(x2-lH(x) = G(x) Then the corresponding approximate critical values can be computed for any significance level a, by solving the equation G(x) = a. Of course, this equation has to be solved iteratively (by the Newton-Raphson method, say). Since exact and asymptotic moments are available, C3 and c^ can be computed likewise. Note that from (3.9) and (3.10), c3 ~ 0(l/n) and c^ ~ 0(l/n). If we choose to do the two-term approximation, then we would be neglecting ct,, and hence neglecting terms of 0(l/n). Therefore, c3 can be approximated by terms whose orders are less than 1/n. In this case, 6c3 ~ /27n (2+ (1A)> , and the approximation becomes F(x) ~*Cx) - a/6)/(^{2+(lA)}(x2-l)<>(x). In the case of a three-term approximation, we would be neglec ting C5. The fifth moment is unavailable, but we might anticipate 3/2 that c5 ~ OCl/n ' .). If this is the case, then up to the order of neglected terms, 6c3.~ /r?7n1 (2+(lA) > 24c, ~ (2/n){6+(12A) + (1A2) >. 42 and the approximation becomes FCx) ~ »(x) - (Cl/6/[27nl{2+(lA)}(x2-l) + (l/24){C2/n)[6+(.12A) + ClA2)3(x3-3x)}(i>(x). 4.2 DISCUSSION Since the results with the asymptotic moments were virtually the same as those with the exact moments Cas it was with Pearson curves), we only display the table for the two- and three- term fits with exact moments.(see Tables 6CA-D) and Tables 7(A-D) ). We can immediately see that'the series expansion has improved the normal approximation. We point out some interesting results arising from the comparison of the two series approximation. First, while the lower critical values from the three-moment fit tend to be -too conservative, those from the four-moment fit are slightly liberal. This appears to be the case of a > 0.05. In general, the four-moment fit seems to be adequate at the lower tail, except for the usual cases of concern n £ 20 and x <• 3. On the other hand, the upper critical values from the three-moment fit tend to be adequate for most cases, but the inclusion of the fourth moment has made the upper critical values very conservative. The four-moment fit is only satisfactory for n a 50 and \ > 3. Obviously, the Gram-Charlier approximation is not recommended since the much simpler x2 approximation is even better. However, it is interesting to note that a three-term partial sum approximation of the true density of I improves the normal approximation consider ably. The critical values obtained from this approximation may be 43 GRAM-CHARLIER THREE-MOMENT FIT (EXACT) Table 6A (a = O.Ol) X 1 3 5 8 n KL I>U KL IMJ KL IMJ KL I>U | 10 109 140 32 125 : 22 132 26 106 20 •; 105 123 79 115 ] 66 101 66 106 1 50 I 101 93 96 98 86 98 1; 90 94 100 85 96 89 74 . 88 69 84 76 I Table 6B ] (a = 0.05) L 3 X 5 8 n KL IMJ • KL IMJ KL IMJ KL IMJ 1 10 190 381 176 364 ' 192 352 • 199 347 1 20 : 305 370 304 354 . j • 287 • 344 284 330 I 50 ; 354 363 358 384, 1 345 363 ••• 352 ' '345 I-100 384 377 i . 356 370 . 359 362 368 354 1 44 Table 6C (a = 0.10) X 1 3 5 8 n j KL I>U KL IMJ KL I>U KL IMJ 10 587 551 482 626 ; 497 628 515 626 E 20 537 678 659 687 638' 709 : 633 680 B 50 668 726 723 731 683 721 727 720 100 735 733 : 707 727 678 713 703 714 1 Table 6D (a = 0.20) 1 3 X 5 8 n KL IMJ : KL I>U IMJ KL I>U 9 10 974 1083 - 1104 1291 • 1240 1328 ; 1248 1312 B 20 1297 1350 : 1352 1426 .. : 1402 1404 1389 1448 • 50 1517 1471 ' 1436 1472, ; 1430 1450 '• 1439 1453 100 • 1468 1455 1432 1442 1398 1485 1438 1486 3 GRAM-CHARLIER FOUR-MOMENT FIT (EXACT) Table 7A (.'<*= O.Ol) X 1 3 5 8 n 1 KL I>U 1 KL I>U 1 KL I>U I KL IMJ J 10 I 109 81 89 0 . 89 1 5 70 1 20 39' 74 36 82 ; 30 : 32 77 I 50 54 69 62 83 59 83 12 115 100 1 59 74 1 72 62 71 63 . 70 67 . 1 table'7B (a = .0.05) X 1 3 5 8 • n KL I>U 1 KL IMJ KL IMJ | KL I>U 8 10 587 212 305 262 275 259 267 264 20 401 246 317 293 • 303 284 293 283 50 358 303 350 355 , i;: 341 335 I 228 429 100 . 377 342 | 349 359 I 358 344 I 365 342 46 Table 7C (• o = 0.10) X 13 5 8 n , KL I>U I<L I>U j I>U \ Ul I>U 10 968 426 774 504 • 738 502 732 515 20 839 532 761 655 707 663 7 7 639 50 744 687 746 725 • 713 713 747 718 100 759 733 720 725 695 712 709 713 1 Table 7D (a = 0.20) 1 X 3 5 8 n I<L I>U I<L I>U I<L I>U I I<L I I>U 10 2147 2233 1824 1733 1667 1695 1665 1630 20 1672 1678 1619 1587 1600 1579 1 , 1580 1589 50 1628 1598 1538 1554 I I 1520 1522 j 1508 1515 100 1536 1499 1499 1483 J 1429 1514 1469 1504 47 found in Tables A5 - A3. Other types of series expansions could also be examined. Two of the more common ones are Edgeworth expansions, which are known to be equivalent to the Gram-Charlier series of type A, and Fisher-Cornish expansions, which are derived from Edgeworth expansions. Treatment of these can be found in Kendall and Stuart (1958, Vol. 1, pp. 157 -157). 48 5. THE LIKELIHOOD RATIO AND GOODNESS-OF-FIT TESTS In the previous chapters, we examined various approximations to the distribution of the index of dispersion for the case that the data is distributed as Poisson in order to obtain approximate critical values. We compared the performances of critical values obtained from large sample approximations, series expansions and Pearson curve fits, and found that Pearson curves seemed to give the most accurate critical values. The one remaining question we will attempt to answer is: "How good is the test based on the index of dispersion relative to other tests of the null hypothesis that the data is distributed as Poisson?" Two well-known methods of testing the adequacy of the mo.del under the null are (i) The Likelihood Ratio Test and O'i) Pearson's Goodness-of-Fit (GOF). To assess the performance of the test based on the index of dispersion, we can examine the power of these three tests against appropriate alternatives. In testing for over-dispersion, ecologists have used the negative binomial (Fisher, 1941), Nermann's conta gious distribution Type A (Neymann, 1939) and Thomas' double Poisson (Thomas, 1949) as alternatives to the Poisson distribu tion. P. Robinson (1954) has pointed out that the Neymann distri bution may have several modes (leading to non-unique estimates when estimating by maximum likelihood) and that a basic assumption 49 of the double Poisson may not be satisfied by the distribution of plant populations. The negative binomial distribution is perhaps the most widely applied alternative to the Poisson. Letting the parameters of the negative binomial be k and 9 (k>0,e>0), we may write: where x=0,l,2,... . From this, we have that E(X) = ke and Var(X) = ke(.l+e) = E(X).(.l+e) > E(.X). For alternatives involving under-dispersion, we can test the null against the positive binomial (although it will 6e noted that the maximum likelihood estimator of n, the number of Bernoulli trials, may not be unique). 5.1 THE LIKELIHOOD RATIO TEST Let r± = (k,e) be a two-dimensional vector of parameters and let f(x,_n) be the probability mass function of the negative binomial. It is shown in Appendix A1.4 that as k + » and e 0 in such a way that ke =X, a constant, then the limiting distribution arrived at is the Poisson with parameter X which has probability mass function f(x,x). Let 0O and 0 be the space of values that the parameters X and n_ may take on, respectively. The GOF problem then is to test H0: n e 0O Hj: n_ e 9-90 50 The likelihood ratio statistic for testing H0 against Hi is A = sup Lfx.nJ/sup* L(x,n), where L(x.,n) is the likelihood function for a sample Xj»...,X . Here, "sup" indicates a supremum taken over QQ while "sup" " indicates a supremum taken over e. Note that this implies thatA<l. In general, the distribution of the likelihood ratio statistic is unknown. However, under regularity conditions (Kendall and Stuart (1958), vol. 1, pp. 230-231), as n -> °>, it is known that asymptotically where p and q are the dimensionalities of the parameter spaces under the alternative and the null, respectively. We now compute the MLE's of X,k and 9. The likelihood functions of the Poisson and negative binomial are respectively, -2 ln A * x2 p-q L(x,X) n xi -x n x 1 e A/x ' i=l 1 n ' = xx* e"nV n x.'. , and i=l 1 L(x,k,e) = n 1 / (e/U+ejr'WU+e)} i=l\ x. / (5.1) Let l0(>^,x) and Ij^.k.e) be the corresponding log-likelihood functions. Then n l0(x,X) = x. InX -nX - X In(x.l) i=l 1 (5.2) 51 But Mx.k.e) = £ in{(k+x.-l)! / Cx '. (k-1)!]} + x. lne 1=1 1 1 - (x. + nk) In (1+6) = E 1 n {(k+x.-l)! / (k-1)!} - Z In (x.l) i=l 1 1=1 1 + x. In e - (x.+nk) In (1+e). z In {(k+x,-l):/(k-l)'.} i=l 1 = C £ + E ] .In {(k+x.-l): / (k-1).} {i:x.=0} {i:x.>0} 1 Since the summation over the zero values of x. is zero, n + E ln{(k+x,-l)l/Ck-l):} =.E ln{Ck+x,-l):/(k-l):}, 1=1 1 i 1 where E+ denotes a summation over i such that x.>0. This sum can be i written as, z+ zMnCk+j-l), i j=l 52 and hence, liU.k.e) = x.lne - (x.+nk)ln(l+e)+ E E1 ln(.k+j-l) i J=l n - 2 ln(x'). (5.3) i=l 1 To obtain the MLE of X, (5.2) must be maximized with respect to x while the MLE's of e and k are obtained by maximizing (5.3) with respect to 6 and k simultaneously. Thus, 3lQ/8X = (x./x) - n . Setting this derivative to zero.and solving for X yields X , the MLE of X as, X = X. (.5.4) Similarly, ali/ae = (x./e) -{(x.+nk)/(l+e)} ali/ak = E E1 {l/Ck+j-1)} - n ln(l+e), i j=l Setting the derivatives to zero, the first equation can be explicitly solved for e to yield 6 = X/k. " (5.5) 53 Substituting this value into the second equation leads to E+ Z1 U/Oj-1)} - n lnU+(X7k)} = 0. (5.6) i j=l Levin and Reeds (1978) give a necessary and sufficient condition for the uniqueness of the MLE of k. This criterion can be stated as: "k, the MLE of k, exists uniquely in CO,00) if and only if n n n EX2- E X. > ( E X.)2/n. (5.7) i=l 1 i=l 1 i=l 1 The right-hand side of this criterion is simply nX2, and so (5.7) can be rewritten as n n E (X.-X)2 > E X. , or i=l 1 i=l 1 (1/n) z (X.-X)2 > X. i=l 1 Since S2 = {l/(n-l)> E (X.-X)2 > (1/n) E (X.-X)2 i=l 1 i=l 1 provided that E CX.-X) >0, a consequence of Levin and Reed's cri-i=l 1 terion is that a unique k exists in C0,«) if the index of dispersion is greater than 1. Thus, subject to Levin and Reeds' criterion, the solution to (5.6) can be obtained numerically. Once k, the MLE of k, is obtained, .... A substitution into (5.5) yields e, the MLE of 9. (The case where the cri-terion is not satisfied corresponds to k=», and is discussed in more 54 detail in section 5.4.) Continuing with the likelihood ratio test, we have AAA In A. = l0(x, X) - li(.x,k,e) x. = x'.{(ln k)-l} + (x.+nk')ln{l+(X/k)} - r+ I1 In (k+j-1). i j=l This is the form of the likelihood ratio test. The asymptotic result is that as n -> °° -2 ln A ~ x2.' 5.2 PEARSON'S G00DNESS;0F-FIT TEST A test that assesses goodness-of-fit is the well-known x2 test that was proposed by Karl Pearson (1900). The GOF statistic is: X2 = f (n.-x.)2/X., where n. is the number of times that the integer j is observed in a 3 sample and x. = nPCX=j) where X ~ P(x), is the expected number of times 3 the integer j will occur under the null. The asymptotic result is that as n -> », v2 ss Y2 * Vl where v is the number of cells. (Note that one degree of freedom is lost since the probabilities computed uner the null are subject to the constraint that they sum up to 1. Also, in the simulation that follows, the value of X is specified and hence no further degree of freedom is lost.) This approximation has been known to work well particularly if the expected number of observations, X., in each cell is 55 at least 5. Now, for sample sizes of about 10 to 20, this rule of thumb may not always be satisfied. The rule that has been implemented is that the expected number of observations in each cell is at least 3. As will be seen, the x2 approximation was still satisfactory in this case. 5.3 POWER COMPUTATIONS We now have three tests whose power we wish to compare. Since the index of dispersion and the test based on X2 do not depend on explicit alternative hypotheses, we might expect the likelihood ratio test to be superior of the three. Because of computational difficulties that may arise when using the likelihood ratio test (these are mentioned later on in this section), it is not recommended for use in practice. We use it here only to provide a baseline for the assessment of the power of the test based on the index of dispersion. Since the index of dispersion is devised to test for the variance being different from the mean, it will be geared towards alternative hypotheses which have this property and so we might expect the index of dispersion to perform better than the test based on X2. Note that while a one-sided test was implemented for the test based oh the index of dispersion (and necessarily for the likeli hood ratio test), the test based on X2 is necessarily two-sided. This should be taken into consideration when comparing the power of the tests. Let us recall the hypotheses we are testing: Ho* Xj,X2»...,X ~ P(.X) HI: x1,x2,...,xn ~ NB(k,e). Through simulation studies, we are going to compare the power of the three tests. However, as the null hypothesis does not specify a par ticular value of x, it is not clear how to choose k and 9 for the 56 simulation. To do this, we argue as follows: Since the Poisson distribution with parameter x is a limiting case of the negative binomial with parameters k and 9, we can specify X and choose k and 9 so that ke = X. Now we also want to choose k so that the tests exhibit reasonable power. For instance, we do not wish to generate data that yields power that is very close to 1. We would like to choose values of k so that the range of the power covers the unit interval [0,1]. To get a good idea of what k should roughly be, we can examine the asymptotic power of the index of dispersion test. To do this, we need the first four moments of the negative binomial which can be found in Kendall and Stuart (1958, vol. 1, p. 131). Letting v, v2,v3, and vt, denote the central moments of the negative binomial, we have v = ke, v2 = ke(e+l), v3 = ke(e+l)(2e+l), and vh = ke(e+l)Cl+6e+6e2+3ke+3ke2). Substituting these moments into equation (2.6) in Chapter 2, we have that I ~ N(l+e , (l/n)v2) where v2 = 2(1+6)2 + (l+e)(2+3e)/k. Notice that by setting k=X/e-and letting e + 0, we obtain the aysmptotic null distribution of the 57 index of dispersion for the Poisson case, namely I « N(l,2/n). This result was seen in Chapter 2 where the performance of the asymptotic normal critical values was assessed. Hence the asypmtotic power of I can be computed from the set of hypotheses: H :9 = 0 o Hj. :9 = 9j > 0 . Let u(e) = 1+e and o2(e) = .(l/n)v2, where v2 is defined as above. If we let I = 1+z /(.2/n), where z is the upper critical value a a a of the standard normal distribution at significance level a, then the asymptotic power of I is Power = *(-wa), where Wq = Ua-y C^i )}//oi5(e1). The asymptotic power of I is presented in table 8 for the case n = 20 and a = 0.05. Table 8: ASYMPTOTIC POWER OF THE INDEX OF DISPERSION TEST (n = 20, a = 0.05) k x = ke 3 5 7 • 10 SIZE 1 .352 .224 .166 .125 .05 3 .739 .560 .425 .309 .05 5 .873 .752 .663 .484 .05 58 Thus, the values of k = 3,5,7 and 10 seem to be adequate. Notice the pattern in this table. The power decreases with increasing k and decreasing e. This is not surprising because as ik-increases and e decreases, the negative binomial approaches the Poisson, and hence it would be much harder to detect differences between the null and the alternative with k large and e small. We now proceed with the Monte Carlo simulation. A total of 500 samples of n = 10, 20 and 50 negative binomial random variables were generated for k = 3, 5, 7 and 10. The three statistics , -2 InA, I and X2 were computed using the negative binomial data. While the computation of I and X2 are very easy on the computer, some problems may occur in computing the likelihood ratio statistic, as mentioned previously. First, the computation of the double.sum in (5.6) at each iteration will increase the cost of running the computer program. This will be more evident for large n and/or large X. Second, for some of the samples, a negative value of k was obtained at some point in the iteration process. This may create a problem in computing In {l+(X/k)} in (5.6). Barring all difficulties however, the Newton-Raphson Algorithm achieved convergence in about 5 or 6 iterations. Continuing with the simulation, an attempt is made to treat each test as equal as possible by using x2 critical values in each case. However, a problem may still occur in the power comparison. Since all these tests are based on asymptotic critical values, the asymptotic approximations may not treat each of the three tests exactly the same. 59 For example, for a given sample size, it may be that X is better approximated by x2 than I, which in turn may be better approximated by X2 than -2 In A. This may make the conclusions on the power comparison unreliable. Proceeding with the power computations, the number of statistics which fell in the rejection region were counted for each of the three tests (Recall that a one-sided rejection region was formed for the tests based on A and I while a two-sided rejection region is necessary for X2). The power of each test is displayed in tables 9-10 (A-D). Each cell in this table contains the power of the likelihood ratio test, the index of dispersion test and the GOF test, in that order. To provide a handle on the accuracy of the x2 approximation for each of the three tests, the estimated size of each test is also displayed in each table. If the approx imation were good for a particular test, then the estimated size of that test should be close to the specified significance level. As mentioned above, the x2 approximation may hot treat these three tests equally. This in fact is the case when n = 10. The critical values for the likelihood ratio test are too conservative as can be seen from the estimated size of the test. For example,, when a = 0.05, the estimated size of the likelihood ratio test is slightly less than 0.01. On the other hand, the estimated size of the test based on X2 is very close to the true significance level for all a, while the test based on the index of dispersion tends to be intermediate. Thus we could infer that if exact 60 POWER OF TESTS BASED ON A.I AND X2 fn=10) Table 9A: a = 0.01 k X = ke 3 5 7 10 SIZE .028 .016 .012 .008 0 1 .068 .044 .032 .028 .008 .010 .008 .006 .006 .008 .148 .058 .026 .018 0 3 .252 .138 .086 .056 .002 .156 .096 .086 .072 .010 .356 .148 .086 .042 .002 5 .478 .276 .180 .114 .004 .290 .192 .128 .102. .024 Table 9B: ct = 0.05 k . X = k9 3 5 7 10 SIZE .064 .048 .038 .032 .008 1 .162 .114 .092 .074 .034 .062 .068 .060 .054 .055 .270 .144 .098 .058 0 3 .444 .272 .224 .152 .038 .242 .174 .136 .112 .050 .486 .286 .182 .122 .006 5 .648 .464 .332 .252 .036 .388 .268 .208 .166 .068 61 Table 9C: a = 0.10 . • k A=l<e 3 5 7 10 SIZE .118 .066 . .052 .048 .020 I .222 .174 .130 .110 .046 .086 .096 .092 .088 .094 .344 .198 .150 .098 .012 3 .566 .384 .310 .242 .070 .272 .204 .156 .136 .090 .570 .752 .250 .178 .006 5 .752 .588 .448 .342 .082 .516 .384 .312 .264 .136 Table 9D: a =0.20 k X=k.9 3 5 7 10 SIZE .166 .108 .086 .072 .036 1 .366 .294 .256 .236 .124 .224 .206 .200 .196 .182 .452 .286 .232 .178 .040 3 .688 .546 .466 .398 .158 .424 .332 .256 .244 .184 .656 .470 .356 .258 .040 5 .848 .718 .618 .504 .160 .582 .456 .382 .326 .216 62 POWER OF TESTS'BASED 0*1 • A , T ANH y2 (n=?n) * Table 10A: a = 0.01 k  X=k9 3 5 7 10 SIZE .044 .016 .012 .010 0 1 .102 .050 .034 .024 .010 .048 .038 .030 .030 .016 .314 .142 .076 .034 .002 3 .450 .228 ,150 .098 .012 .266 .136 .096 .062 .024 .664 .334 .190 .112 0 5 .770 .472 .312 .190 .008 .558 .296 .202 .140 .022 Table 103: a = 0.05 k X=kO 3 5 7 10 SIZE .104 .050 .042 .032 .010 1 .214 .140 .100 .082 .046 .078 .056 .052 .050 .040 .502 .258 .164 .104 .012 3 .666 .416 .298 .218 .032 .410 .266 .206 .156 .056 .804 .534 .342 .208 .008 5 .898 .706 .526 .366 .042 .686 .408 .314 .224 .074 63 Table IOC: a = 0.10 k X=ke 3 5 7 10 SIZE .172 .104 .076 .054 .020 1 .316 .228 .174 .148 .086 .148 .128 .110 .104 .098 .594 .336 .232 .158 .020 3 .780 .544 .420 .320 .086 .498 .328 .268 .210 .110 .872 .620 .452 .290 .020 5 .936 .796 .650 .486 .086 .760 .514 .408 .298 .112 Table 10D: -a = 0.20 x=ke 10 SIZE .246 .492 .226 .164 .388 .202 ,124 ,328 ,174 ,104 .272 .174 .054 .172 .166 .704 .460 .326 .246 .032 3 .882 .714 .604 .476 .196 .614 .462 .382 .308 .194 .916 .746 .578 .396 .046 5 .966 .886 .792 .662 .192 .834 .652 .542 .422 .220 64 critical values were employed, the power of the likelihood ratio test would be considerably larger than indicated in the tables. Similarly, the critical values for the index of dispersion test are slightly conservative and hence we would expect the power of this test to increase if exact critical values were used. The power of 2 the test based on X , however, would be pretty much what the tables indicate. Turning to n = 20, the same problem still arises for the likelihood ratio -- very conservative critical values. On the other hand, while the asymptotic critical values used for the index of dispersion are still slightly conservative, the approximation has clearly imprPved and the estimated size of the index of dispersion test is closer to the true significance level. In fact, the estimated power of the index of dispersion is close indeed to its asymptotic power. Although it is not clear that the likelihood ratio test is more powerful than the test based on the index of dispersion, we can make one additional observation if we compare tables with the same size (for instance, the 20% table for the 1 ikel ihood ratio and the 5% table for the index of dispersion when n = 20), we see that in each cell, the estimated power of the likelihood ratio test is indeed higher than that of the index of dispersion - however, only marginally. This is an indication of what we might expect to see if the sample size were large enough so that the estimated size' of the test is close to the true significance level. 65 Thus, the results displayed in tables 9 and 10 seem to suggest the following order in terms of the power of each test. Likelihood Ratio, Index of Dispersion and GOF based on X2. A further attempt to compare the power of the likelihood ratio and the index of dispersion, tests are displayed in table 11 (A-D) for a sample size of n = 50. As before, we may compare the 20% table for the likelihood ratio with the 5% table for the index of dispersion to conclude that the likelihood ratio test is only slightly more powerful than the test based on the index of dispersion. 5.4 THE LIKELIHOOD RATIO TEST REVISITED At the time when this thesis Was first being written, no obvious explanation could be made about the conservatism of the critical values of the likelihood ratio test. Subsequently, the explanation became clear: For the situation under consideration, the null distribution of -2 1 n A does not converge to that of a x2> but rather to that of a mix ture of a x2 and a zero random variable, each with probability 1/2. This is an example of the general results of Chernoff (1954). The reasoning goes as follows: A A A If the MLE (6,k) for the negative binomial occurs at k=°°, then A-l and -2 In A= 0. Levin and Reeds (1977) have established that this occurs if and only if (n-l)S s nX. Thus, under the null hypothesis, we have P(-21n A H 0) = P{(n-l)S2 <; nX> = P{n(.S2-X) - S2 s 0} = P(/nC(S2-x) - CX-X)3 - U/^)S2 * 0} = P(/ii[(S2-x) - (X-x)]^0>, for large n. 66 POWER OF TESTS BASED ON A AND I (n=50) Table 1'IA: a = O.Ol k X=k8 3 5 7 10 SIZE .094 .034 .016 .010 .004 .166 .068 .050 .032 .012 .744 .376 • .172 .082 .002 .842 .510 .304 .152 .010 .984 .762 .500 .270 .002 .990 .844 ' .636 .400 .008 Table 113: a = 0.05 X =ke 10 SIZE ,224 ,354 .098 .222 .060 .156 .034 .106 .014 .046 ,886 .936 .572 .724 .374 .532 ,206 .354 .018 .046 .996 .998 .890 .950 .714 .818 .*52 .622 .014 .040 67 Table 11C: ct = 0.10 k X=k8 3 5 7 10 SIZE .314 .178 .120 .080 .028 .522 .318 .234 .178 .106 .922 .668 .482 .296 .032 .960 .810 .642 .484 .096 .998 .942 .778 .564 .034 .998 .978 * .896 .722 .098 Table 11D: cc = 0.20 X=ke 10 SIZE .430 .664 .950 .984 .268 .434 .770 .910 .198 .380 .600 .784 .152 .308 .422 .638 .066 .216 .076 .192 .998 .968 .876 .676 .070 1.000 .988 .956 .848 .188 68 N(Q,E), where E = From sections 2.1 and 2.2, we have that 7r7CX-x}" "x x _x x +-2x£ Letting f(x,y) = y-x so that f (X,S2) = S2-X = (S2-x) - (X-x), we have, as a consequence of the delta method, that /h"{(S2-x) - (X-X)}-^ N(0,2X2). Thus for large n, we have that P(-2 In A = 0) » 1/2; i.e. -2 In A = 0 approximately half the time. Thus under the null, we have the following result: -2 In A ( 0 with probability 1/2 X2 with probability 1/2 (5.8) As a supplement to (5.8), the first 500 Poisson samples from the 15,000 previously generated were again used in order to check if half of these 500 samples would give a value of -2 In A = 0. Table 12 displays the number of samples out of. the'500 which led to -2 In A = 0. Table 12: Number of Times (n-l)S2 n X 10 20 50 1 367 337 304 3 340 317 302 5 338 ' 307 293 69 The fact that the entries in the table decrease as ngets large is indeed encouraging and re-affirms our position that the null distri bution of -2 InA converges to a mixture of distributions. What effect then does (5.8) have on the power computations? In the previous computations, we have been assuming that a = P(-2 1n A s C ), a where Ca is the upper critical value corresponding to a x2 distribu tion. Letting Z be a standard normal random variable, we have instead that P{-2 ln A s CM * (l/2)P{I0>Ca}+ (1/2)P{Z2>C> = (1/2){1-P[Z2<C ]} a = 1-*(7C'-), a where I0 = 0 with probability 1 and $ is the standard normal CDF. But,-a = P(Z2>Ca) = 1 - P(-/C < Z < /C ) a a = 2{l-<»(/Ca) , and hence, P(-2 ln A s C ) « a/2, a instead of the anticipated value of a'. Further simulations were not done as enough information can be gathered from the previous results. In particular, using the correct asymptotic critical values for the likelihood ratio test, for each fixed n, the previous results for a = 0.10 are the appropriate results for a = 0.05 and the previous results for a = 0.20 are the appropriate results for a = 0.10. These are displayed in Tables 13,14 and 15 (A-B). 70 POWER OF TESTS BASED ON A,I AND X2 (n=10) Table 13A: a = 0.05 k x=ke 3 5 7 10 SIZE .118 .066 .052 .048 .020 l. .162 .114 .092 .074 .034 .062 .068 .060 .054 .056 .344 .198 .150 .098 .012 3 .444 .272 .224 .152 .038 .242 .174 .136 .112 .050 .570 .752 .250 .178 .006 5 .648 .464 .332 .252 .036 .388 .268 , .208 .166 • .068 Table 13B: a = 0.10 k i X=k9 3 5 7 10 SIZE .166 .108 .086 .072 .036 1 .222 .174 .130 .110 .046 .086 .096 .092 .088 .094 .452 .286 .232 .178 .040 3 .566 .384 .310 .242 .070 .272 .204 .156 .136 .090 .656 .470 .356 .258 .040 3 .752 .588 .448 .342 .082 .516 .384 .312 .264 .136 71 POWER OF TESTS BASED ON A,I AND X2 (n=20) Table 14A: a = 0.05 k  X=ke 3 5 7 10 SIZE .172 .104 .076 .054 .020 1 .214 .140 .100 .082 .046 .078 .056 .052 .050 .040 .594 .336 .232 .158 .020 3 .666 .416 .298 .218 .032 .410 .266 .206 .156 .056 .872 .620 .452 .290 .020 5 .898 .706 .526 .366 .042 .686 .408 .314 .224 .074 Table 14B: a = 0.10 k X=k8 3 5 7 10 SIZE .246 .164 .124 .104 .054 1 .316 .228 .174 .148 .086 .148 .128 .110 .104 .098 .704 .460 .326 .246 .032 3 .780 .544 .420 .320 .086 .498 .328 .268 .210 .110 .916 .745 .578 • .396 .046 5 .936 .796 .650 .486 .086 .760 .514 .408 .298 .112 72 POWER OF TESTS BASED ON A AND I (n=50) Table 15A: a = 0.05 k X=ke 3 5 7 10 SIZE 1 .314 .354 .178 .222 .120 .156 .080 .106 .028 .046 3 .922 .936 .668 .724 .482 .532 .296 .354 .032 .046 5 .998 .998 .942 .950 .778 .818 .564 .622 .034 .040 Table 15B: a = 0.10 k X=k8 3 5 7 10 SIZE 1 .430 .522 .268 .318 .198 .234 .152 .178 .066 .106 • 3 .950 .960 .770 .810 .600 .642 .422 .484 .076 .096 5 .998 .998 .968 .978 .876 .896 .676 : .722 .070 .098 73 The correction to the asymptotic null distribution of -2 ln A has certainly created a better picture. The estimated size of the likeli hood ratio test is closer to the nominal significance level than it was when the x2 approximation was employed. However, the critical values l for the likelihood ratio test are still very conservative. Hence, the power of the likelihood ratio test would be greater than that displayed in these tables. As before, we may compare tables with approximately the same estimated size. For example the power of the test based on the index of dispersion from Table 13A might be compared to the power of the likelihood ratio test from Table 13B and similarly, Table 14A to 14B. We see that for n=10 and 20, the likelihood ratio test is only margin ally better than the test based on the index of dispersion. For the case n=50, no reasonable comparison can be made, but we would expect the same behavior from both tests. 74 6. CONCLUSIONS As mentioned in Chapter 1, the index of dispersion is a statis tic often used to detect departures from randomness. As the null dis tribution of the index of dispersion is unknown, large sample approxi mations were used as a preliminary fit. The asymptotic null distribu tion of I was seen to be normal with mean 1 and variance 2/n. Asymp totic critical values from this distribution were then employed and assessed by a Monte Carlo simulation. The results were that the nor mal approximation was very poor for sample sizes typically encountered in practice and that this approximation only becomes satisfactory for a sample size of about 100 and x > 5. A further attempt to improve the normal approximation was made by an infinite series expansion of of the true null distribution of I,. We saw that a three-moment fit from the Gram-Charlier expansion improved the normal approximation enormously, but that this approximation was only satisfactory for n _ 50. The x2 approximation on the other hand seemed to be fairly accurate for n>20 and X>3. This is certainly encouraging because of one important reason - the x2 approximation is simple to apply. To further improve the x2 approximation (particularly for the cases n<20 and;x<3), Pearson curves were utilized. We found that except for the case n=10 and X=l, Pearson curves definitely improved the approximation. 75 Two issues still remain unanswered: (i) What should be done in the case n=10 and X=l? (ii) How well will the approximations remain when we A replace X by X=X? For the second question, we expect that the Pearson curve approxima tion will still perform well. As for the first question, let us keep in mind the suggestion put forth by Fisher (1950) and Cochran (.1936) — that the test based on the index of dispersion should be carried out conditionally, particularly when the Poisson parameter X is small, for then exact frequencies can be computed. Finally, the comparison of the powers of the tests based on the likelihood ratio, the index of dispersion and Pearson's X2 statistic showed that the test based on the index of dispersion exhibits reason able power when the hypothesis of randomness is tested against over-dispersion. This supplements the results obtained by Perry and Mead (.1979). From the basis of accurate critical values and reasonably high power, we conclude that the index of dispersion is highly recommend-able for its use in applications. 76 REFERENCES ANDERSON, T.W. (1958). "An Introduction to Multivariate Statistical Analysis". Wiley, New York. BATEMAN, G.I. (1950). "The Power of the x2 Index of Dispersion Test When Neyman's Contagious Distribution is the Alternative Hypothesis". Biometrika, 37, 59-63. BLACKMAN, G.E. (1935). "A Study by Statistical Methods of the Distribution of Species in Grassland Communities". Annals of Botany, N.S., 49, 749-777. CHERNOFF, H. (1954). "On the Distribution of the Likelihood Ratio". Annals of Mathematical Statistics, 25, 573-578. CLAPHAM, A.R. (1936). "Over-dispersion in Grassland Communities and the Use of Statistical Methods in Ecology". Journal of Ecology, 24, 232-251. COCHRAN, W.G. (1936). "The x2 Distribution for the Binomial and Poisson Series with Small Expectations". Annals of Eugenics, 7, 207-217. CRAMER, H. (1926). "On Some Classes of Series Used in Mathematical Statistics". Skandinairske Matematikercongres, Copenhagen. CRAMER, H. (1946). "Methods of Statistics". Princeton University Press, Princeton. DARWIN, J.H. (1957). "The Power of the Poisson Index of Dispersion". Biomterika, 44, 286-289. DAVID, F.N. and MOORE, P.G. (1954). "Notes on Contagious Distributions in Plant Populations". Annals of Botany, N.S., 28, 47-53. ELDERTON, W.P. and JOHNSON, N.L. (1969). "Systems of Frequency Curves". Cambridge University. FISHER, R.A. (1941). "The Negative Binomial Distribution". Annals of Eugenics , 11, 182-187. FISHER, R.A. (1950). "The Significance of Deviations from Expectation in a Poisson Series". Biometrics, 6, 17-24. FISHER, R.A., THORNTON, H.G. and MACKENZIE, W.A. (1922). "The Accuracy of the Plating Method of Estimating Bacterial Populations". Annals of Applied Biology, 9, 325. GRAD, A. and SOLOMON, H. (1955). "Distribution of Quadratic Forms and Some Applications". Annals of Mathematical Statistics, 26, 464-477. 77 GREEN, R.H. (1966). "Measurement of Non-Randomness in Spatial Distributions". Res. Popul. Ecol., 8, 1-7. HALDANE, J.B.S. (1937). "The Exact Value and Moments of the Distribution of x2, Used as a Test of Goodness-of-Fit, When Expectations are Small". Biometrika, 29, 133-143. HALDANE, J.B.S. (1939). "The Mean and Variance of x2, When Used as a Test of Homogeneity, When Expectations are Small". Biometrika, 31, 419-355. HOADLEY, A.B. (1968). "Use of the Pearson Densities for Approximating a Skew Density Whose Left Terminal and First Three Moments are Known". Biometrika, 55, 559-563. HOEL, P.G. (1943). "On Indices of Dispersion". Annals of Mathematical Statistics, 14, 155-162. KATHIRGAMATAMBY, N. (1953). "Notes on the Poisson Index of Dispersion". Biometrika, 40, 225-228. KENDALL, M.G. and STUART, A. (1958). "The Advanced Theory of Statistics". Vol. 1. Griffin, London. KENDALL, M.G. and STUART, A. (1958). "The Advanced Theory of Statistics", Vol. 2. Griffin, London. LANCASTER, H.O. (1952). "Statistical Control of Counting Experiments". Biometrika, 39, 419-422. LEVIN, B. and REEDS, J. (1977). "Compound Multinomial Likelihood Functions are Unimodal: Proof of a Conjecture of I.J. Good". Annals of Statistics, 5, 79-87. MENDENHALL and SCHEAFFER (1973). "Mathematical Statistics with Applications". Duxbury Press, North Scituate, Massachusetts. MULLER, P.H. and VAHL, H. (1976). "Pearson's System of Frequency Curves Whose Left Boundary and First Three Moments are Known". Biometrika, 54, 649-656. NEYMAN, J. (1939). "On a New Class of Contagious Distributions Applicable in Entomology and Bacteriology". Annals of Mathematical Statistics, 10, 35-57. PEARSON, E.S. and HARTLEY, H.O. (1966). "Biometrika Tables for Statisticians", Vol. 1, 3rd ed., Cambridge. 78 PEARSON, K. (1900). "On a Criterion that a Given System of Deviation from the Probable in the Case of a Correlated System of Variable is Such that it can be Reasonably Supposed to Have Risen in Random Sampling", Phil. Mag., (5), 50, 157 PEARSON, K. (1901). "Systematic Fitting of Curves to Observations". Biometrika, 1, 265. PERRY, J.N. and MEAD, R. (1979). "On the Power of the Index of Dispersion Test to Detect Spatial Pattern". Biometrics, 35, 613-622. POLYA, G. (1930). "Sur Quelques Points de la Theorie des Probabilites". Ann. de L'lnst. Henri Poincare, 1, 117-162. ROBINSON, P. (1954). "The Distribution of Plant Populations". Annals of Botany, N.S. 18, 35-45. SOLOMON, H. (1960). "Distributions of Quadratic Forms - Tables and Applications". Technical Report No. 45, Applied Mathematics and Statistics Laboratories, Stanford University, Stanford, California. SOLOMON, H. and STEPHENS, M.A. (1978). "Approximations to Density Functions Using Pearson Curves"; Journal of the American Statistical . Association, 73, 153-160. STUDENT (1919). "An Explanation of Deviation from Poisson1s Law in Practice". Biometrika, 12, 211-215. THOMAS, M. (1949). "A Generalization of Poisson's Binomial Limit for Use in Ecology". Biomtrika, 36, 18. 79 APPENDIX Al.l THE CONDITIONAL DISTRIBUTION OF A POISSON SAMPLE GIVEN THE TOTAL Let Xj,...,X be independent identically distributed Poisson random variables with parameter X. Then the sum of the X.'s n 1 X • — E X • ) i=l 1 is distributed as Poisson with parameter nX. Consider the joint distribution of X^,...,Xn given the total X.. Since fx|x.(-) = fx^)/fx.(x-) = ( n xVVx ')/{CnX)x-e"nV(x.):} i=l = (x.:)(l/n)x7 n x.l , the desired result follows, i.e. (xx ,...,Xn|X.) ~ Mult (.X.,1/n,1/n,1/n,...1/n). The distribution of a vector of independent and identically distributed Poisson random variables conditioned on the total is a 1 multinomial with parameter m = X. and equal cell probabilities 1/n. This conditional distribution is independent of the Poisson parameter X since X. is a sufficient statistic forx . The moment generating funtion of the multinomial is {PiexpUi) + ... + pnexp(tn)lm. In our case this becomes M(t) = {(l/n)[exp(t!) + ... + exp(tn)]}X'. 80 A1.2 THE FIRST FOUR MOMENTS OF I From [3.5), we see that we require the evaluation for x.>0 of E{(S2/x)k |x.=x.}, for k=l,2,3 and 4. Now for k=l and x.>0, we have n (n-l)X-E{S2/X|X.=x.} = E{ z (X .-X)2 | X.=x.l i=l 1 n = E{ z X 2 - nX2|X.=x.} i=l 1 n = Z E{X.2|X.=x.} - nX2 i=l 1 = n{Var(X.|X.=x.) + E2(X. |X.=x.)}-nX2 = n{x.(l/n)[l-(l/n)]+(x./n)2> - nX2 = (n-l)X . It follows that for x.>0, we have E{S2/X | X.=x.} = 1. (A1.3) For k=2 and x.>0, we begin by noting that {(n-l)sY= ( ^(X.-X)2)2 = • Z (X.-X)1* +.z z (X.-X)2(X.-X)2. i=l 1 lVj 1 3 Upon expanding these powers of (X^-XJ and evaluating the required conditional expectations through the moment generating function of (X^.X^, ... ,X )|X., we have, for x.>0, that E{(S2/X)2 jx.=x.l = (n+l)/(n-l) - (2/nCn-l))(1/X). CA1.4) It follows that for x.>0 Var(S2/X|X.) = {2/(n-l)}{l-(_/nX)}. 81 Considerably more algebraic effort is required in the cases k = 3 and 4. For x.>0, we obtain E{(S2/X}3|X.=x.} = (n+l)Cn+3)/Cn-l)2 - 2{l/(n-l)2}-{[l+Cl3/n}]Cl/X) + (2/n)[l-(6/n)](.l/X2)} (A1.5) Ef/(S2/xT|X.=x.} = (n+l)Cn+3)Cn+5)/Cn-l)3 + {2/(n-l)3>-{2n[l+C5/n)][l-(17/n)]Cl/X} - (.2/n2)(2n2+ 53n - 261)(1/X2) - (4/n3l(.n2- 30n +90)(1/X3)}. CA1.6) Equations (Al.3)-(Al.6) agree with those provided by Haldane (.1937). Substitution of these conditional moments into (3.5) yields exact expressions for the first four raw moments of I which are given • in (3.7). To obtain the central moments from these raw moments is a matter of using the formulas given in Kendall and Stuart (1958, vol. 1, p. 56). We should mention that the algebra involved in computing these conditional expectation was checked by UBC's symbolic manipula tor, documented in "UBC REDUCE". Expansions of powers and the compu tation of the partial derivatives of the moment generating function were all checked on the computer. It remains to evaluate E+(l/X^), for j =1,2 and 3. Now, E+(.l/X) = nE+(l/X.) -fl k = n £ (l/k)e e /k! , where 9=nX. k=l 82 If we let then or f(e) = E (l/k)e'6ek/k:, k=l co f(e) = -f(e) + E e"8ek_1/k! k=l f'(e) + f(e) = e"e(ee-l)/e. Since the solution to this differential equation is f(e) = e~8/8 {et-l)/t)dt, o it follows that E+(l/X) = ne"9/6 {(e^D/tldt. o Simi1arly, E+(l/X2) = n2(e"eln e/8 [(et-l)/t]dt o - e~V (In t)[(et-l)/t]dt>, and E+(l/X3) = n3C(l/2)e*6(ln e)2 fQi{et-l)/t}dt 0 - e"8ln 6 fQ (In t){ (e^D/tldt o + (l/2)e"8 /8 (In t)2{(et-l)/t)dt]. o None of the above integrals can be evaluated explicitly, and P2' .V31 and ]ik' would either have to be approximated by numerical integration or by an asymptotic expansion. We illustrate this by e panding the integrals for large 6. Let (l/n)E+(l/X) = f(6), (l/n2)E+(l/X2) = g(6) and (l/n3)E+(l/X3) = h(6). 83 Under the transformation t=9x, we have f(6) = e'8 /{(e8x-l)/x}dx o l l = e"8ln x (e8x-l)| - e"8 / (In x)ee8xdx l f 0 = -ee"9 / ln(l-z)e8^1_z^dz, where z=l-x. Now, and so o For j = 1,2,3,..., let •ln(l-z) = z + z2/2 + z3/3 + f'(e) = / (z + z2/2 + z3/3 + ...)ee"0zd: Ue) = / zjee"8Zdz J o 3 -ez,1 , * J-l -ez. = -z e + J / z e dz. o o We then have an expression for Ij in terms of II., k<j: I .(e) = -e'8 + (j/e)I ._1(e), where I0(e) = 1 - e"8. For j^l, this recursive formula yields I j (e) = -e-8 + (j/e){-e~8 + _(j-l)/e_l (e)} "fl - (j/e-)e"e• + {j(j-l)/82}{-e-e+C(j-2)/e]l .(e)} -e = -e = j'./eJ + 0(e-8) ~ j I/ej Therefore, f(e) ~ _ k'./ek+1 + 0(l/eN+2) as N k=0 and the asymptotic expansion for E+(l/X) is E+(l/X) ~ n(l/e + l/e2 + 2/e3 +...). 84 Notice that the first term approximation of E+(l/X) is n/e = 1/E(X), which would be the naive approximation to this expectation. The asymptotic expansions for g(e) and h(e) are obtained in a similar fashion, except instead of expanding ln(l-z), we would need to expand (ln(l-z)l2 and (In(l-z)}3 for g(e) and h(e) respectively. The results are E+(l/X2) ~ n2(l/e2 + 3/03 + ll/e4 + ...), E+(l/X3) ~ n3(l/63 + 6/e* + 35/65 + ...). Alternately, these same expansions could be obtained by repeated applications of L'Hospital's rule. Substituting these expansions into (3.7) yields the raw moments, correct to 0(l/nt*): Pi' ~ 1 , vi' ~ 1 + (2/n) + (2/n2)[ 1 - (1/x)] + (2/n3)[ .1 - (1/x) -(1/X2)] + (2/n-)C 1 - (1A) - (1A2) - (2/x3)] , y3' ~ 1 + (6/n) + (2/n2)[ 7 - (1A)1 + (?/n3)[ 11 - (15A) - (3/X2)] + (2/nMC 15 - (29/x) - (7A2) - (8A3)] , yit' ~ 1 + (12/n) + (4/n2)[ 14 + (l/x)1] + (4/n3)[ 37 - (9/x) - (1A2)] + (4/nMC 72 - (115A) -.(68/x2) - (6A3)1 . (A1.7) 85 Proceeding in the same way as Hoel. (1943), we can also assess the accuracy of the x2 approximation to the null distribution of I by examining the ratio of the asymptotic moments of I with the moments of [l/(n-l)lx2 The behavior of these ratios as n and/orX increases, will indicate when the x2 approximation is satisfactory. The first four moments of a random variable distributed as [l/(n-l)lx2 ^ are: V = 1 . oiz' = (n+l)/(n-l) , 103' = (n+l)(n+3)/(n-l)2 , cV = (n+l)(n+3)(n+5)/(n-l)3 , (see Mendenhall and Scheaffer, 1973, p.138). Notice that the moments of Cl/(n-l)lx2 j approximate the moments of a Mult (x.,1/n...,1/n), correct to 0(l/n). Let Ri = Vj'/uif'. for i = 1,2,3 and 4 Cnote that R^l for all n and'X). Using the asymptotic expressions in (A1.7), these ratios are computed for n = 10,20,50 and 100 and x = 1,3,5 and 8, and entered in Table Al. The asymptotic moments of the index of dispersion agree very well with those of the x2_j distribution for n^20 and X^l. In fact, this is also apparent for n^lO and X^ 5. As n and/or X increases, R2,R3, and R^ all approach the limiting value 1. This is indeed encouraging and compliments the results obtained in section 2.5. 86 TABLE Al: The Ratios of the Moments of I and x2 i Cfor each cell, the ratios ^2*^3 and R4 are entered in that order) X n 1 3 5 8 0.9797 0.9937 0.9962 0.9977 10 0.9631 0.9887 0.9932 0.9957 0.9724 0.9921 ' 0.9948 0.9962 0.9950 0.9984 0.9990 0.9994 20 0.9925 0.9976 0.9986 0.9991 1.0003 1.0002 1.0001 1.0001 0.9992 0.9997 0.9998 0.9999 50 0.9991 0.9997 0.9998 0.9998 1.0009 1.0003 1.0002 1.0001 0.9998 0.9999 0.99996 0.99993 100 0.9998 0.9999 0.99998 0.99998 1.0003 1.0001 1.00004 0.99996 87 Al.'3 THE TYPE VI PEARSON CURVE We rewrite the differential equation given by (3.1) as dClog f(x)}/dx = Cx-al/IbzCx-Aj)Cx-A2)}, (A1.8) where Ai and A2 are the roots of the quadratic b0 + bxx + b2x2. Kendall and Stuart (1958, vol. I,p.l49) give the formulas for a,b0, bx and b2 as functions of 8i,B2 and y2. When using these formulas, one should keep in mind that the formulas were obtained assuming the origin at the mean. For the type VI case, both roots of the quadratic are real and have the same sign. Without loss of generality, assume that A2>Ai. Then, by partial fractions, we can write dllog f(x)}/dx = U/bzUCi/Cx-A!) + C2/(x-A2)}, where C2 = (a-Ax )/(A2-Ai) = (a-A^/S, C2 = (A2-a)/(A2-A!) = (A2-a)/c> and 5 = A2-Ai. For x>A2, we can integrate equation (A1.8) with respect to x to get log f(x) = (C1/b2)log(.x-A1) + (C2/b2)log(x-A2) + C where C is the arbitrary constant of integration. Transforming back to the true origin, i.e. replacing x by x-1, yields log f(x) = (A/bzJlogfx-ai) + (C2/b2)log(x-a2) + C where ai = 1+Ai and a2 = 1+A2, and hence -Pi Q2 f(x) = k(x-a!) (x-a2) (A1.9) where qi = -Ci/b2, q2 = C2/b2 and k is a normalizing constant. Since A2 > Aj (and hence a2 > aj) and qj and q2 are real numbers, it follows that type VI Pearson curve defined in (A1.9) is 88 a distribution defined on [a2,»). If we let y = x-al , then -qi q2 fly) = ky (y+a^a,,) , for y^-a^O, -qi Q2 = ky (y-s) , since 6 = A2 - Ai = a2 - ai Now let z = S/y so that dy/dz = -6/z2. Then f(y) = kC«/z) qiCC5/z)-5]q2|dy/dz| q2-qi+l qi-2 q2 = kfi z [Cl-z)/z] qi-q2-2 q2 = k'z U-z) , for 0<z<l. This last form of the density of the beta distribution is what is required when using the IMSL library to compute critical values. A1.4 A LIMITING CASE OF THE NEGATIVE BINOMIAL The negative binomial distribution with parameters k and 8 approaches different distributions depending on the limiting operation. In particular, let k -> <» and e -> 0 in such a way that ke = x, a constant. If X ~ NB(k,e), then the moment generating function of X is Mx(t) - {p/(l-qet)}k, where p = l/(e+l) and q = e/(e+l). Hence, 89 Mx(t) = {Cl/(e+l)]/[l-(e/9+l)et]}k = {[k/(X+k)][l-(x/(A+k))et]}k = (k/CxCl-e*) + k]}k = {l+[X(l-et)/k]}"k (A1.5) But as k ->• », the limit of the right-hand side of (A1.5) is gX(e -1)^ w(11-cn js preCiSely the moment of generating function of the Poisson distribution. 90 A2.1 HISTOGRAMS OF I ro II II c tv> OJ to o o o CD o I— CO I-H zc CM in in • - 2 > • o (U O H O I-I- > l/> IX UJ ty> co O O m Z ai • < • UJ o _ i-Z O </> 3 O h-O O 2 (J — UJ l/> UJ -I oc o a 00 x x oc > o m _ I o < UJ • i3 2 < Z3 •r- — o o _ CO CO •— CM co in CO r~ UJ o » O N Tf (J) ID —• CO "» UJ Z —• co t- CO o — ai a. •H — >• o X o n co CM CO 01 o CO t-2 _ o o — CO co UJ u — CM CO in CO t~ O • UJ 1- o tN 10 10 —• CO *r CC Z to r~ 03 o — u. *~* « * # < • • o + X X X X X X co I X X X X X X 1 X X X X X X 1 X X X X X X I X X X X X X in + X X X X X X t- I X X X X X X X 1 X X X X X X X 1 X X X X X X X 1 X X X X X X X o X X X X X X X I X X X X X X X 1 X X X X X X X 1 X X X X X X X 1 X X X X X X X in X X X X X X X CO 1 X X X X X X X I X X X X X X X 1 X X X X X X X I X X X X X X X O X X X X X X X ID 1 X X X X X X X 1 X X X X X X X 1 X X X X X X X 1 X X X X X X X m X X X X X X X in 1 X X X X X X X 1 X X X X X X X I X X X X X X X j X X X X X X X O X X X X X X X in t X X X X X X X 1 X X X X X X X 1 X X X X X X X 1 X X X X X X X in X X X X X X X 1 X X X X X X X 1 X X X X X X X 1 X X X X X X X I X X X X X X X O X X X X X X X <T 1 X X X X X X X 1 X X X X X X X 1 X X X X X X X 1 X X X X X X X in X X X X X X X o 1 X X X X X X X 1 X X X X X X X t X X X X X X X X 1 X X X X X X X X O X X X X X X X X co 1 X X X X X X X X 1 X X X X X X X X 1 X X X X X X X X 1 X X X X X X X X in X X X X X X X X CM 1 X X X X X X X X 1 X X X X X X X X 1 X X X X X X X X t X X X X X X X X O X X X X X X X X CN 1 X X X X X X X X 1 X X X X X X X X 1 X X X X X X X X 1 X X X X X X X X in X X X X X X X X 1 X X X X X X X X 1 X X X X X X X X 1 X X X X X X X X 1 X X X X X X X X o X X X X X X X X X *- 1 X X X X X X X X X 1 X X X X X X X X X 1 X X X X X X X X X 1 X X X X X X X X X in X X X X X X X X X 1 X X X X X X X X X 1 X X X X X X X X X 1 X X X X X X X X X 1 X X X X X X X X X 4 •1 + + + + •» + + o o o o o o o o o < o o o o o o o o o O o o o o o o o o O o o o o o o CO o Ul Ul i ID CO o CM «I to o CM 1- *£ tN m IP co a> 2 < 2 « 11 « • « » « « II . i 0) Q O aioo 88 + o CO in r-x x X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X + + o in u> o CO in m O m O 1-O co in CM O CM X X X X X X X X X X X X X X X XX X X X X X X X X x x : + + X X X X X X •i 1 CM ti in to r- co o CMCMtNCMrMCMCMCMfOCOCO FIG. A2 HISTOGRAM OF I (.1000 samples, n = 10, X = 51 SYMBOL COUNT MEAN ST.DEV. X 1000 0.981 0.447 EACH SYMBOL REPRESENTS 1 OBSERVATIONS INTERVAL FREQUENCY PERCENTAGE NAME 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 INT. , CUM. INT. CUM. *.240000 +XXXXXXXXXXXXXX 14 14 1 .4 1 .4 *.360000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 35 49 3 .5 4.9 *.480000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 62 111 6 .2 11.1 *.600000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 101 212 10 . 1 21.2 *.7 20000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 94 306 9 .4 30.6 ».840000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 116 422 11 .6 42.2 *.960000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 103 525 10 .3 52.5 *1.08000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 103 628 10 .3 62.8 • 1.20000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 96 724 9 .6 72.4 *1.32000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 83 807 8. .3 80.7 * 1.44000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 51 858 5 . 1 85.8 -1.56000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 43 901 4 .3 90. 1 * 1.68000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 31 932 3 . 1 93.2 * 1.80000 +XXXXXXXXXXXXXXXXXXXX 20 952 2 .0 95.2 *1.92000 +XXXXXXXXXXX 11 963 1, . 1 96.3 "2.04000 +XXXXXXXXXXXXX 13 976 1, .3 97.6 -2.16000 +XXXXX 5 981 0 .5 98 . 1 -2.28000 +XXXXXXXX 8 989 0. .8 98 .9 '2.40000 +XXXXX 5 994 0. .5 99.4 -2.52000.+XXX 3 997 0. 3 99.7 -2.64000 + 0 997 0. ,0 99.7 -2.76000 + o 997 0. 0 99.7 -2.88000 + XX 2 999 0. .2 99.9 • 3.00000 +x 1 1000 0. 1 100.0 -3.12000 + 0 1000 0. 0 100.0 -3.24000 4- 0 1000 0. 0 100.0 + + + + + + + ._+ + + +— + + + +—--+ 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 FIG. A3 HISTOGRAM OF I (1000 samples, n = 20, X SYMBOL COUNT MEAN ST.DEV. X 1000 0.998 0.330 EACH SYMBOL REPRESENTS 1 OBSERVATIONS INTERVAL FREQUENCY PERCENTAGE NAME 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 INT, . CUM. INT. CUM. *.300000 + 0 0 0 .0 0.0 * . 400000 +XXXXXXXXXXX 11 11 1 . 1 1 . 1 *.500000 +XXXXXXXXXXXXXXXXXXXXXXXX 24 35 2 .4 3.5 *.600000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 64 99 6 .4 9.9 *.700000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 91 190 9 . 1 19.0 *.800000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 103 293 10 .3 29.3 *.900000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 114 407 11 .4 40.7 * 1.00000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX/XXXXXXXXXXXXXXXXXXXXXX* 142 549 14 .2 54 .9 •1.10000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 99 648 9 .9 64.8 * 1.20000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 98 746 9 .8 74.6 * 1.30000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 84 830 8 .4 83.0 * 1.40000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 53 883 5. ,3 88.3 • 1.50000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 55 938 5. .5 93.8 * 1 .60000 +XXXXXXXXXXXXXXXXXXXXXXX 23 961 2 .3 96. 1 • 1.70000 +XXXXXXXXXXX 11 972 1 . , 1 97.2 * 1.80000 +XXXXXXXXXXX 11 983 1 , , 1 98.3 * 1.90000 +XXXX 4 987 0, ,4 98.7 •2.00000 +XXX 3 990 0. ,3 99.0 •2.10000 +XXXX 4 994 0, , 4 99.4 •2.20000 +XXX 3 997 0. ,3 99.7 *2.30000 + 0 997 0. 0 99.7 •2.40000 +XX 2 999 0. 2 99.9 *2.50000 + • 0 999 0. 0 99 .9 "2.60000 +x 1 1000 0. 1 100.0 •2.70000 + 0 1000 0. 0 100.0 •2.80000 + 0 1000 0. 0 100.0 Co 5 10 15 20 25 30 35 40 45 50 5S 60 65 70 75 80 93 LCI II o CM II c l/> CU I a. E ta in o o o < 3 I— CJ z tu CJ • CC H KJ Z 0. >-! > • CJ I Z 3 UJ o o • ui I-o: Z O + to Ont in oo oiinoovMii^OiiKtnouiMogioooo 00'-tOo»-Mr)tn'»»-ij)nii)Mociioio)0)oioooo ^^nviniPr-cococococnrocQCncococnoooo CD O CD CD CM (/) • CO Z > • O UJ O M Q I-< I— > CO CX. LU CO m ro o O) 2 0)"-< • UJ O Z O in 3 O I-o o z (J *- ui CO ul _i o: o a. co CU > 10 _l o m s. >-CO I o < > a UI UJ H- E 2 < >-• Z X X X + + O O o o o o o o o o CM co CO *- (orocs^rocMCMOiiooo CO (0 co CM o CM 0) I- co CM •r-*~ in co 0) in CO o r- 0) O 10 T o co CM CO in cn CO in r» *~ CO in CD t> co CO OJ 0) 0) CO to CO CM •o: CO CM CM 0) u> 00 CO 10 to CN o CM 0) r> f- ro CM *~ « « « « « « X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X + + + + + + + + + + + + + 4 O oo in O in io o CO in in O in in O in CO o CO in CM O CM X X X X xxx xxx xxx xxx xxx X X X X X X X + + + + + + + + oogooooooooooooooooooooo ooooooogoooooooooooooooo OOOOOOOOOOOOOOOOOOOOOOOO oooooooooooooooooooooooo OOOOOOO^cMCO^inior~coo)0^(Nco^inu)t~ •3ini0f-coo> '"'---''-'-"-"-"-"-••-OICMCMCMCMCMrMCM KoointltjiKaiitjiitlitajit)..!. FIG. A5 HISTOGRAM OF I (.1000 samples, n = 50, \ = 3) SYMBOL COUNT MEAN ST.DEV. X 1000 1.003 0.204 EACH SYMBOL REPRESENTS 1 OBSERVATIONS INTERVAL FREQUENCY PERCENTAGE NAME 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 INT CUM. INT. CUM. •.550000 +XXXX 4 4 0 4 0 4 *.600000 +XXXXXX 6 10 0 6 1 0 •.650000 +XXXXXXXXXXXXXXX 15 25 1 5 2 5 •.700000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 35 60 3 5 6 0 •.750000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 35 95 3 5 . 9 5 •.800000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 48 143 4 8 14 3 •.850000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 67 210 6 7 21 0 •.900000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 109 319 10 9 31 9 •.950000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 1 13 432 11 3 43 2 •1.00000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 105 537 10 5 53 '7 • 1.05000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 94 631 9 4 63 1 •1.10000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 77 708 . 7 7 70 8 •1.15000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 75 783 7 5 78 3 • 1.20000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 67 850 6 7 85 0 •1.25000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 42 892 4 2 89 2 •1.30000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXX 28 920 2 8 92 0 •1.35000 +XXXXXXXXXXXXXXXXXXXXXXX 23 943 2 3 94 3 • 1 .40000 +XXXXXXXXXXXXXXXX 16 959 1 6 95 9 •1.45000 +XXXXXXXXXXXXX 13 972 1 3 97 2 • 1.50000 +XXXXXXXXXX 10 982 1 0 98 2 •1.55000 ,+xxxx 4 986 0 4 98 6 * 1.60000 +XXXX 4 990 0 4 99 0 •1.65000 +XXXX 4 994 0 4 99 4 •1.70000 +XX 2 996 0 2 99 6 -1.75000 +XXX 3 999 0 3 99 9 •1.80000 +X 1 1000 0. 1 100 0 DO + + + + + + + + + + + + + j, + + + 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 I_<_OO-J.-~IOIOIUIJ> ID(0Offlv|U|lnilulMJOMM^OOolOI>l»W<nOt.O CDMono bOOM<no^coM(jiOJ>cofooOOOOOOOO OOOOOOOOOOOO OO oooooooooooo oooooooooooooooooooooooooo oooooooooooooooooooooooooo > z S -1 m m JO < > + + + X + + X X X X X X X X o to CO o co o Ol Ul O Ul Ul CD O cn ui O ui oa O + + • X X X X X X X X X X X X X X X X X X X X X X X X X X X + + X X X X X X •o + X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X t + X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X + • X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X . . XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX X XX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX XXX X X X X X X X X X X X X X X X X X X X X X X X X X X X + + + + X X X X x x x.x XXX XXX XXX XXX xxx XXX xxx xxx xxx xxx xxx xxx X X X X X X X X X X X X X X X X X X X X X X X X : x : x : x X X X X X X X . . X X X X X X X X XX X X X X X X X X X X X X X X X X X X X X 00 + o > n I </i •< 3 CO o 1— in -< JJ X s m 00 "O o JO r~ m </i m — r> Z Q o -1 5 c l/i o z H — o o o — CO (/) to 71 < > -I o O m O • < Z KJ • l/l O to CD Ol —I O CD 73 !> _ O — o o o to CD to 3 II cn o v >» II cn — — N>_UI<J>~J_MK3 — comeo — — 00-»0 — J>4*00O'-JUl--l0T>Ul_O — M — m—~JJ>0!K>0 O Q O i O O O i 10 00 CD -J 01 Ul J> CO CO (0 CO (0 01 o M O CO -J M (0 CO 03 01 OOOOOOOO — — K><OUIC1~4CJMM— (OOICO — — OO oo->o-fc*o)o>i oi s moiuo-M-m->iJ>oiMO OOQ<oiO(0(o_io__<r>a>oo~j.oiuiJ>_K>-— OOOID(0IC(fllHO~IUl(J(0QNl(0fflbM-'-'0IU-OO O O Ol0_COi>OMKJCJlO<0--JIO(O(000alUllD00 — -aioo *~> ~n Z JO -I m • o c O m c z 2 O • -< z -—4 JO • o m z n H c > _ a • m 96 FIG. A7 HISTOGRAM OF I (.1000 samples, n = 100, \ = 3)_ SYMBOL COUNT MEAN ST.DEV. X 1000 1.001 0.144 EACH SYMBOL REPRESENTS 1 OBSERVATIONS INTERVAL FREQUENCY PERCENTAGE NAME 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 BO INT. CUM. INT. CUM. -+ • .'640000 + XXX •.680000 +XX •.72OOO0 +XXXXXXXXX •.760000 +XXXXXXXXXXXXXXXXXXXX *.80OOO0 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX •.840000 +-XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX •.830000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX •.920000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* •.960000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* • 1.00000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXjft(XXXXXXXXXXXXXXXXXXXXXXXXXX* • 1.04000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* • 1.08000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* •1.12000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX • 1.16000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX •1.20000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX •1.24000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX •1.28000 +XXXXXXXXXXXXXXXXXXX •1.32000 +XXXXXXXXXXXXX •1.36000 +XXXXXXXXXXXX •1.40000 +XXXXXX •1.44000 +XX »1.48000*+XXX -1.52000 + •1.55000 + • 1.60000 +X • 1.64000 + + + + + + + 4. + 4. + + + + 4. 4. 4. 4. 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 3 3 0. .3 0 .3 2 5 0. .2 0 .5 9 14 0. 9 1 .4 20 34 2. ,0 3 .4 39 73 3. 9 7 .3 59 132 5. 9 13 .2 66 198 6. 6 19, .8 105 303 10. 5 30, .3 98 401 9. 8 40. . 1 124 525 12. 4 52, .5 108 633 10. 8 63. .3 85 718 8. 5 71 . .8 79 797 7. 9 79. .7 75 872 7. 5 87. .2 39 911 3. 9 91 . . 1 33 944 3. 3 94, .4 19 963 1 . 9 96. .3 13 976 1 . 3 97, .6 12 988 1 . 2 98. .8 6 994 0. 6 99. .4 2 996 0. 2 99. .6 3 999 0. 3 99. g 0 999 0. 0 99. .9 0 999 0. 0 99. 9 1 1000 0. 1 100. ,0 0 1000 0. 0 100. 0 FIG. A8 HISTOGRAM OF I (.1000 samples, n = 100, X = 5) SYMBOL COUNT MEAN ST.DEV. X 1000 1.000 0.141 EACH SYMBOL REPRESENTS 1 OBSERVATIONS INTERVAL FREQUENCY PERCENTAGE NAME 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 INT, , CUM. INT. CUM. *.500000 + • 0 0 0 .0 0 .0 *.550000 + 0 0 0 .0 0 .0 *.600000 + X 1 1 0 . 1 0, . 1 *.650000 +x 1 2 0 . 1 0 .2 *.700000 +XXXXXXXX 8 10 0 .8 1 .0 •.750000 +XXXXXXXXXXXXXXXX 16 26 1 .6 2, .6 •.800000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 42 68 4 .2 6. .8 •.850000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 75 143 7 .5 14, .3 •.900000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 107 250 10, ,7 25. 0 •.950000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 129 379 12 .9 37. .9 • 1.OOOOO +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 152 531 15, .2 53. . 1 • 1.050O0 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 1 18 649 11. .8 64. .9 •1.1OOOO +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX• 123 772 12. .3 77. .2 •1.15000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX* 82 854 8. .2 85. 4 •1.20000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 57 911 5. ,7 91 . 1 •1.25000 +XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 48 959 4 . 8 95. 9 •1.30000 +XXXXXXXXXXXXXXXXX 17 976 1. .7 97. .6 •1.35000 +XXXXXXXXXXXXXXXX 16 992 1. .6 • 99.. ,2 • 1.40000 + XXXX 4 996 0. ,4 99. 6 •1 .45000 +x 1 997 0. 1 99. 7 * 1 .50000 + XX 2 999 0. 2 99. 9 • 1.55000 + 0 999 0. ,0 99. 9 • 1 .60000 +x 1 1000 0. 1 100. ,0 •1 .65000 + 0 1000 0. 0 100. 0 - 1 .70000 + 0 1000 0. 0 100. 0 • 1 . 75000 + 0 1000 0. 0 100. 0 5 10 15 20 25 . 30 35 40 45 50 55 60 65 70 75 80 98 FIG. A.9: NORMAL PROBABILITY PLOT FOR I (100C Samples, n = 10, X = 3) -3.7S • •*••..*....*....*....•....*....*....*....•....•....*....•....*... * • * * .250 .750 1.25 1.75 2.252.753.253.75 " 0.00 .500 1.00 1.50 2.00 2.50 3.00 3.50 4.00 99 FIG. A.10: NORMAL PROBABILITY PLOT FOR i (1000 Samples, n = 10, X = 5) .+....+. -3.75 * • .•.... + .... + ...+....+,.. . + .... + ..,. + .... + .... + ....•.... + .. ..+....+... • • 20 .60 1.0 1.4 1.8 2.2 2.6 3.0 0.0 .40 .80 1.2 1.6 2.0 2.4 2.8 3.2 100 FIG. A.11: NORMAL PROBABILITY PLOT FOR I (10C0 Samples, n = 20, \ = 3) -3.75 • .... + .... + .... + ....*.... + .... + .... + .... + .... + .... + .... + ....•.... + .... + .... + ....* .450 .750 1.05 1.35 1.65 1.95 2.25 2.55 .300 . 600 . 900 1.20 1.50 1.80 2.10 2.40 101 FIG. A.12: NORMAL PROBABILITY PLOT FOR I (100C Samples, n = 20, \ = 5) • * * * • • * * • • • • • • • • • • * • * * * .3750 .6250 .8750 1.125 1.375 1.625 1.875 2.125 2500 .5000 .7500 1.OOO 1.250 1.50O 1.750 2.OOO 102 FIG. A.13: NORMAL PROBABILITY PLOT FOR I (1000 Samples, n = 50, X = 3) 3.00 2.25 X 1 .50 P E C T E .750 0 N 0 R 0.00 M A L V -.750 A L U E -1 .50 • • • • • * • * • • * • * • • • • • « » * * • • • • * -2.25 -3.00 -3.75 • .»....*....*....*....•....•....•....•....•....•....*....•....•....•....•....•.., .450 .630 .810 .990 1.17 1.3S 1.53 1.71 .540 .720 .900 1.08 1.26 1.44 1.62 1.80 103 3.75 * FIG. A.14: NORMAL PROBABILITY PLOT FOR I (1C00 Samples, n = 50, X = 5) * * • * * * * • * -3.75 + .630 .810 .990 1.17 1.39 1.53 1.71 1.89 .540 .720 .900 1.08 1.26 1.44 1.62 1.80 104 3.75 • FIG. A.15: NORMAL PROBABILITY PLOT FOR I (10CC Samples, n = ICO, X = 3) • • a * • * -3.75 + ... + .... + ....•.... + .... + ....•.... + .... + .... + ..:. + .... + .... + ....•*•.... + ... . + .... + ... .5250 .6750 .8250 .9750 1.125 1.275 1.425 1.575 .6000 .7500 .9000 1.050 1.200 1.350 1.500 1.650 105 FIG. A.16: NORMAL PROBABILITY PLOT FOR I (10CC Samples, n = ICO, X = 5) -3.75 * .5250 .6750 .8250 .9750 1.125 1.275 1425 i S7S 6O00 .7500 .9000 1.050 1.200 1.350 1.500 ' 1 106 A2.2 EMPIRICAL CRITICAL VALUES CM r-x O CTl CM CM o CO O CM CO CO o LO CM IT) O CO CO r-x o o CO 1—1 LO CTl 0x1 CM o oo r—1 cn oo co CO CO r-x r^ CO 1—1 1—1 r-H o CM o CTl o cn UD co CO LO o o o o CO CO CO CO «xf CO «xT • • • • • • • • • • • • • • • • • CM CM CM CM CM CM CM CM r-H 1—1 I-H 1—1 r—l r-H I-H t—1 i—1 co CM CTl CO r---3" CTl CM i—l CM oo -3- "xT "xT LO i—1 i—1 CO oo i—1 o CM CO CM xj-CTl CO o CO LO r-x i—1 i—I o CTl LO CO CM CM x* "xt CO CM o o CTl CTl cn i—1 r—1 I—1 o r-x r-x r-x r-x •xT -xT «3- <xt" CO CO CM CM • • • • • • • • • • • • • • • • CM CM CM CM I-H 1—1 1—1 T—1 1—1 r-H 1—1 I-H I—l 1—1 t—1 CO 1 QL in CO co o CM LO 1—1 CO CTl oo co CO CO •xT o r-x CTl cn I—l LO 1—1 CM CO 1— CO LO r-x. LO CTl LO CO CM LO co LO r-x CO 00 CTl CTl oo CO LO LO LO x3- x* x3" CO oo 00 co oo LO LO LO LO CO CO CO CO CM CM CM CM • • • • • • • • • • • • • • •• • • o i—i I-I 1—1 i-H 1—1 T—1 1—1 1—1 1—1 I-H 1—1 1—1 I-H 1—1 T—t o o c IO 1—4 CD CO r-x o CO CM *3" oo cn CO CM CTl CTl cn r-x CO CXJ r-x CO LO CO CO CM CO CTl rx. CO LO CO CO CO CO zr o CTl t~t CM CM CTl CO CO CO CO CO CO CO 00 00 oo co o cn LO CO CO CO CO «xt- «xt CM CM CM CM I—1 I-H 1—1 I—l • • • • • • • a • • • • • • « • • 1  i i—1 r-H i—1 i—l 1—1 I—1 f—1 I-H r-i r-H i-H I—l I-H T—1 T—t I—l CO =C CO o CTl CO CM 1—1 -3- oc CM CO cn 1—1 CO CO 1— I-H o O cn CO cn co LO r-H CM CO CO CO CO CO i—i o o r-x CO CO xj- 1—1 1—1 1—1 LO LO LO LO CM CM CM CM i—i LO co CO CO CO f-x.rx.rx. fx. 00 00 00 00 LO • • • • • • • • • • • • • • • • • CO CO UJ •xt r-- LO oo CO o CTl co r-x o CTl LO CTl CO CTl => CM LO CO r-x. CO LO CM CO LO r-x I-H CO I—l CO o _J IT) CO rx. CO CO CO CO CO o CTl CTl CTl 00 oo 00 00 <: O CO CO CO LO LO LO LO r-x CO CO CO r-x r-x r-x r-x > • • • • " • • • • i •a: c_> t—t CO CO LO CO co l-x r-x LO «3- oo CO LO r-x CTl CO CO 1— LO I—l CTl CO LO CO r-~ o 00 CO CO CO "xT CM I—l •—I CM CO o CTl CTl CTl r-x r-x r-x LO "xt x3- xfr -3" or O CO ro CM CM «3- "xT CO CO CO co r- l-x, l-x. r-. c_> • • • • • • • • • • • * • • • _i C_) DC cn CM r—i LO co LO CTl i—l «3" CM CO cn fxx LO CM oo o l—l o CM co i—l co CO CTl co oo CO r-x CO x3- co CM 1—1 T—t O- o CM cn CTl 00 o CO CO co • r-x LO CO LO 00 r-x r-x r-x s: • CM t—i i—l 1—1 CO CO LO LO CO LO CO CO CO CO LU • • * • • CM i •a: i—l CO LO CO I—1 co LO 00 1—1 CO LO CO r-t co LO 00 UJ i o o o Q TAE c r-i CM LO o t—1 107 A2-3 PEARSON CURVE CRITICAL VALUES ID in r-l ri o rH r-l cn o r- co vo ro VD r-l in 1— rH o rH Oi IO ro ro cn in rr -r CO rH o o Ol VD VO VO VD o o o o VD VD vo vo • • • • • • • • • • CN CN CN CN CN CN CN CN rH rH rH rH in CN O r~ r~ r» in VD rH Ol VO CN O in CO CN rH O O VD in r- r- O o o ro ro ro ro ro ro ro CT> o rH rH rH r-r-• • • • • • • * • • • • * CN CN CN CN rH rH rH rH rH rH rH rH CN cn CN VO CO ro rH in r- •ST rH in in cn i-» CN CN in VD -r -3* -r CN in VD r- r- CO co CO in in in in cn CO CO CO CO in in in in ro ro ro ro • • • • t • • • • • • • • rH rH rH rH rH rH rH rH rH rH rH rH o CO CO in ro in ro O -1* r- CN VO VD O CO ro rH >> cn CN in m o r~ rH rH CN rH CN CN CN VD VD VO VO cn in VO VO VD -r CN CN CN CN • • • • • • • • • • • • • rH HHH rH rH rH rH rH rH rH rH a 1— ro ro CO rH CO in cn VO ro vo o o -i' CO CN CO CO rH CO VO o M' ro CN «c rH r-l r~ VD ro CN rH rH VD in in in X • m VD VD VO VO • LU • » • • • • • • • • • • *— CO 11 J _ o CN o r- CO ro o CO cn rH rH _J r--ro r-•3' CN CN CO VO VD VD in rji •< in CN CO r~ r-» VO ro ro o cn oi cn > o ro ro ro in in m in vo vo vo • • • • • • • • ' • • • • • _ C_) 1— i—i —1" ro o VD CO ro ro o _ in cn O in CN rH CO -r CN CN cn r- VD CM in rH o o O r~ r- VO -r o ro ro ro ro m vo vo VO vo UJ • • • • • • • • • • • • • la ce: _D CJ z: o in cn in VD in cn ro ro r- o *r o m VO o CO cn rH ro CN rH CN o oo CO o cn CO CO CTl VO vo vo 00 VO vo in cx o CN rH rH rH ro ro ro ro in in in in •< • • • • • • • • • • • • • o_ ro H ro in co rH ro in co H ro in co LE o o o CO rH CN in •< 1— ro CO CN r-O CO VD CN rH O o 1 • • • rH rH rH rH cn VO r~ rH CN cn CO CO o cn cn cn ro CN CN CN « • • • rH rH rH rH ••tf VO CN O vo in m in -r -?•-(• TI« CN CN CN CN • • • • rH rH rH rH vo vo r-- cn rr in in in co co co co rH rH rH rH • • • • H H rH rH r- rH vo CN vo -r co ro CN CN CN' CM CO CO CO CO cn in vo rH -i' o cn oi co co r-» r» »> r> ro o oo H CO TJ- CN CN ^J* rj4 r- r- r- r-~ H rl rf tf H vO M co » r- r-vo vo vo vo rH ro in co o o rH TABLE A4: PEARSON CURVE CRITICAL VALUES (ASYMPTOTIC) n X .005 .025 .05 .10 a .90 .95 .975 .995 .1 « I 8 .2169 .1898 .1928 .1972 .3477 .3099 .3070 .3046 .4216 .3832 .3775 .3759 '.5140 .4786 .4720 .4687 1.4443 1.6108 1.6196 1.6259 1.8188 1.8601 1.8692 1.8726 2.0673 2.0998 2.1060 2.1084 2.6656 2.6431 2.6340 2.6275 20 50 1 .3953 Z5038 .5642 .6388 1.4114 1.5753 1.7342 2.0988 3 .3674 .4789 .5424 .6219 1.4244 1.5828 1.7315 2.0573 5 .3642 .4747 .5382 .6184 1.4271 1.5845 1.7307 2.0472 8 .3630 .4728 .5366 .6163 1.4297 1.5848 1.7299 2.0398 1 .5786 .6612 .7062 .7608 1.2621 1.3560 1.4436 1.6341 3 .5627 .6494 .6971 .7548 - 1.2649 1.3545 1.4370 1.6093 5 .5599 .6473 .6951 '.7534 1.2650 1.3545 1.4353 1.6044 8 .5587 .6463 .6945 .7525 1.2653 1.3539 1.4346 1.6014 100 1 .6856 .7503 3 .6763 .7440 5 .6744 .7429 8 .6733 .7420 .7852 .8270 1.1852 .7804 .8240 1.1863 .7796 .8233 1.1860 .7794 .8234 1-1862 1.2474 1.3042 1.4239 1.2459 1.2994 1.4110 1.2455 1.2987 1.4085 1-2447 1-2981 1-4069 •TABLE A5: GRAM-CHARLIER CRITICAL VALUES (THREE EXACT MOMENTS) a 10 X .005 .025 .05 .10 .90 -95. .975 .995 1 .2291 .3078 . .3750 .4711 1.6649 1.9223 2.10.94 2. 4263 3 .1618 .2581 .3357 .4433 1.6708 1.9292 2.1257 2. 4618 5 .1489 .2489 .3286 .4384 1.6720 1.9300 2.1279 2. 4673 8 .1418 .2438 .3247 .4357 1.6727 1.9304 2.1291 2. 4703 20 50 1 .4127 .4855 .5419 .6184 1.4492 1.6200 1.7537 1.9847 3 .3713 .4601 .5238 .6076 1.4459 1.6095 1.7437 1.9812 5 .3626 .4550 .5202 .6055 1.4454 1.6075 1.7415 1.9800 8 .3577 .4521 .5182 .6044 1.4452 1.6064 1.7402 1.9793 1 .5913 .6557 .6989 .7542 1.2722 1.3683 1.4496 1.5974 3 .5694 .6446 .6917 .7505 1.2700 1.3616 1.4407 1.5884 5 .5647 .6423 .6903 .7498 1.2696 1.3603 1.7415 1.9800 8 .5621 .6411 .6894 .7494 1.2694 1.3595 1.4379 1.5852 iob 1 .6924 .7482 3 .6799 .7423 5 .6773 .7411 8 .6758 .7404 .7822 .8243 1.1887 .7786 .8226 1.1875 .7778 .8223 1.1873 .7774 .8221 1.1872 1.2517 1.3065 1.4097 1.2480 1.3010 1.4030 1.2473 1.3000 1.4015 1.2469 1.2994 1.4007 TABLE A6: GRAM-CHARLIER CRITICAL VALUES (THREE ASYMPTOTIC MOMENTS) a n X .005 .025 .05 .10 .90 .95 .975 .995 10 20 1 . 2406 . 3131 .3770 .4698 1.6848 1.9478 2.1346 2.4506 3 . 1940 .2855 .'3598 .4630 1.6496 1.9001 2.0898 2.4139 5 . 1828 .2792 .3559 .4615 1.6433 1.8901 2.0799 2.4054 8 . 1761 .2756 .3537 .4606 1.6399 1.8845 2.0741 2.4005 1 . 4172 .4876 .5429 .6186 1.4524 1.6252 1.7593 1.9901 3 . 3830 .4698 .5322 .6144 1.4387 1.5999 1.7319 1.9653 5 . 3751 .4658 .5299 .6135 1.4362 1.5948 1.7260 1.9597 8 . 3705 .4635 .5285 .6130 1.4349 1.5920 1.7227 1.9565 50 1 . 5924 .6562 3 . 5725 .6471 5 . 5681 .6451 8 . 5656 .6441 :6991 .7542 1.2725 .6938 .7522 1.2683 .6927 .7518 1.2675 .6921 .7516 1.2670 1.3690 1.4505 1.5983 1.3592 1.4379 1.5846 1.3573 1.4353 1.5816 1.3563 1.4338 1.5798 100 1 . 6928 .7483 .7823 .8243 1.1888 1.2519 1.3067 1.4100 3 . 6810 .7432 .7793 .8232 1.1869 1.2472 1.3001 1.4017 5 . 6785 .7421 .7787 .8230 1.1865 1.2463 1.2988 1.3999 8 . 6771 .7415 .7784 .8229 1.1863 1.2458 1.2981 1.3989 TABLE A7: GRAM-CHARLIER CRITICAL VALUES (FOUR EXACT MOMENTS) a 10 X .005 .025 .05 .10 .90 .95 .975 .995 1 .2233 -.3797 '.4598 .5573 1:4398 2.0509 2.2601 2.5756 3 .1123 .2918 \3868 .5010 1.5637 2.0050 2.2406 2.5873 5 .1017 .2764 .3728 .4896 1.5831 1.9941 2.2324 2.5850 8 .0965 .2681 .3651 .4832 1.5930 1.9881 2.2272 2.5829 20 1 .3518 .5031 3 .3266 .4650 5 .3223 .4584 8 .3209 .4548 .5733 .6553 1.3862 .5398 .6289 1.4162 .5336. .6238 1.4208 .5301 .6210 1.4232 1.6631 1.8292 2.0707 1.6260 1.7896 2.0448 1.6202 1.7809 2.0375 1.6172 1.7759 2.0331 50 1 .5616 .6562 .3 .5502 .6439 5 .5481 .6415 8 .4823 .6138 .7058 .7643 1.2592 .6951 .7561 1.2634 .6930 .7545 1.2641 .6919 .7536 1.2645 1.3733 1.4709 1.6321 1.3630 1.4515 1.6110 1.3613 1.4478 1.6061 1.3604 1.4145 1.5670 100 1 .6785 .7474 3 .6715 .7416 5 .6701 .7405 8 .6693 .7398 .7843 .8280 1.1844 .7796 .8246 1.1853 .7787 .8240 1.1854 .7781 .8236 1.1855 1.2524 1.3134 1.4252 1.2482 1.3045 1.4123 1.2474 1.3028 1.4095 1.2470 1.3018 1.4079 TABLE A3: GRAM-CHARLIER CRITICAL VALUES (FOUR ASYMPTOTIC MOMENTS) n X .005 .025 .05 .10 .90 .95 .975 .995 1 .2461 .3181 .3815 • .4737 1.6798 1.9409 2.1263 2.4400 1 n 3 .1651 .2600 .3369 .4438 1.6728 1.9323 2.1288 2.4644 XU 5 .1474 .2481 .3281 .4382 1.6712 1.9287 2.1266 2.4662 8 .1371 .2412 .3231 .4350 1.6703 1.9264 2.1250 2.4668 1 .4181 .4884 .5436 .6191 1.4517 1.6243 1.7582 1.9886 ?n 3 .3724 .4606 .5241 .6078 1.4463 1.6102 1.7445 1.9820 *cu 5 .3621 .4547 .5201 .6055 ' 1.4453 1.6072 1.7411 1.9797 8 .3561 .4513 .5178 .6042 1.4448 1.6055 1.7391 1.9782 1 .5924 .6563 .6992 .7543 1.2725 1.3689 1.450.4 1.5982 50 3 .5696 . .6447 .6918 .7506 1.2701 1.3617 1.4408 1.5886 5 .5646 .6423 .6902 .7498 1.2696 1.3602 1.4388 1.5863 8 .5618 .6409 .6894 .7494 1.2694 1.3594 1.4377 1.5850 100 1 .6928 .7483 3 .6799 .7423 5 .6772 .7411 8 .6757 .7404 .7823 .8243 1.1888 .7786 .8226 1.1875 .7778 .8223 1.1873 .7774 .8221 1.1871 1.2519 1.3067 1.4099 1.2481 1.3011 1.4030 1.2473 1.3000 1.4015 1.2469 1.2994 1.4006 

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