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Saddlepoint approximations to distribution functions Hauschildt, Reimar 1969

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SADDLEPOINT APPROXIMATIONS TO DISTRIBUTION FUNCTIONS by REIMAR HAUSCHILDT .'•' B.Sc, Queen's University •of Kingston, Ontario, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of MATHEMATICS We accept this thesis as conforming to the required standard. The University of British Columbia 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 tha permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of M* rH£Mft-Tt-C$ The University of British Columbia Vancouver 8, Canada Date SePT.'&i*, ii. Supervisor: Dr. J. Zidek ABSTRACT In this thesis we present two approximations to the distribution function of the sum of n independent ran dom variables. They are obtained from generalizations of. asym ptotic expansions derived by Rubin and Zidek who considered the case of chi random variables. These expansions are ob tained from Gurland's inversion formula for the distribution function by using an adaptation of Laplace's method for inte grals. By means of numerical results obtained for a variety of common distributions and small values of n these approxi mations arc compared to the classical methods of Edgeworth and Cramer. Finally, the method is used to obtain approxima tions to the non-central chi-square distribution and to the doubly non-central F-distribution for various cases defined in terms of its parameters. iii. O TABLE OF CONTENTS PAGE INTRODUCTION . 1 • CHAPTER I. . NOTATION AND PRELIMINARY RESULTS 5 1.1 Notation 5 i. 2 ' The Edgeworth Approximation 6 .1.3 The Cramer Approximation 7 1.4 The Saddlepoint Method 10 '.• 1. 5 Remarks 2CHAPTER II. THE SADDLEPOINT APPROXIMATIONS 21 ' • 2.1 • Asymptotic Expansions ' 21 2.2 ..The. Lattice Case • 33 CHAPTER III. COMPUTATIONS 37 3.1' Remarks on the Tables " 37 3-2 " Chi Random Variables;" 39 3.3 The Exponential Probability Law 43 3.4 The Normal Probability Law 49 3.5 ' The Non-Ceritral Chi-Square Probability . 49 Law .... 3.6 The Uniform Probability Law 54 3.7 Remarks .58 CHAPTER IV. OTHER APPLICATIONS 59 • 4.1 . 'The Non-Central Chi-Square Distribution 59 4.2 The Doubly Non-Central F-Distribution 63 : 4.3 Remarks. 70 APPENDIX ' 1  71 Al • Computing N(z) > 71 A2 Computer Program for/Evaluating the 78 Saddlepoint 2 Approximation REFERENCES . •• 86 ACKNOWLEDGMENTS * The author wishes to thank Dr. J. Zidek for sugges ting the topic of this thesis and for generous assistance given during its.writing. The financial support of the National Research Council of Canada and of the University of British Columbia is also gratefully acknowledged. INTRODUCTION . It is often necessary to approximate the distribution of a statistic whose exact distribution is unknown or cannot conveniently be calculated. For example, in the evaluation of error probabilities in some types.of communications systems or sometimes in the determination of the power function of a 1 n likelihood ratio test, the probability P(— £ X. < x) , n i=l 1 where the X^ (i = .1,2,....) are independent random variables, •is required for some values of x 1 A well-known approximation available for this type of problem is due to Edgeworth ([3L pp. 228-229- [7], p. 515). Another is that of Cramer ([7], 'p. 520; [4]), which was designed to provide an accurate approximation even in the case where x is permitted to depend on n When the characteristic function of the statis tic is known, inverting the Fourier transform explicitly is often impossible, while a numerical integration routine is z too time-consuming for procedures requiring high accuracy, especially when U(t)| is not rapidly decreasing as |t| - » . In this thesis we consider two approximations, sug gested by Rubin and Zidek [13], to the distribution function of: the sum of n independent random variables'. Each of these approximations is called, in [13], a saddlepoiht approximation in keeping with the terminology used by Daniels [5], who seems to have been the first to introduce into the literature of probability approximation theory the method upon which the approximations in [13] are based. Whereas in this thesis we are concerned with problems relating to distribution functions, the work of Daniels pertains to density functions. The efforts of Rubin and Zidek [13] are directed toward the problem of finding an approximation to the distri bution function of ( | Z-j^ | + . . . + | Z | ) , where the {Zi} are independent random variables and each Z. (i = l,...,n) is normally distributed with mean 0 and variance 1 Each of their approximations is roughly equivalent, in terms of required computer time, to those of Edgeworth and Cramer. In [13] it is shown for the problem considered there, on the basis of numerical results, that one of.the approximations is superior to either of the two classical methods. Even for the case n = 10 , where for values of the arguments con sidered, the older methods yield an accuracy of at most two significant figures, it gives results accurate to five signi ficant figures. In Chapter I of this thesis all of the approximations mentioned above are presented. .Also given is an inversion for mula, derived from that of Gurland [9], which forms the basis for the saddlepoint approach. The contents of Chapter II consist of proofs that the formal expansions derived in [13] are in fact asymptotic ex- , pansions. In [13] it is suggested that these results might be obtained by using an adaptation of the argument given by Daniels [5], which is based on the method of steepest descent. Here we.give for the case of non-lattice random-variables 3. simple direct proofs which use Laplace's method for integrals and special features of the present problem. The results are, in one case, an expansion in powers of n *- , and, in the other, as in Daniels' case for densities, an expansion in powers of n""*" The approximations are then compared, in Chapter J>, with those of Edgeworth and Cramer for a variety of special cases on the basis of numerical computations for small values of n . The results are qualitatively the same as those found in [13] for the special case considered there, which is presented for completeness in section (3.4). The results given in Chapter 3 are obtained with the intention of comparing the various approximations described j- • earlier. Except for the cases considered in sections 3.4 and 3.7, the desired values can be found with reasonable accuracy from existing tables. In Chapter 4, some additional applica-tions-of the "saddlepoint method"'are considered. • These in volve the non-central chi--square. distribution for large values of its non-centrality parameter,; :and the doubly non-central F - distribution for selected special cases defined in terms of its four parameters. In the latter case particularly, existing tables are inadequate, and the saddlepoint approxi mation may be of practical value. Two appendices are supplied. In the first is given a method of evaluating the normal distribution function for either real or complex values of its argument. The method, which uses continued fractions, is given in [13]. In. Appendix 2, we describe a program written in FORTRAN which is. suitable for numerically evaluating the better, of the two saddlepoint approximations. 5. CHAPTER I NOTATION AND PRELIMINARY RESULTS 1.1 Notation. Let {X-j^Xg,...} be a set of independent, identically distributed random variables, each having distribution function 2 F , density function f , mean u = 0 , variance a , and characteristic function cp ; that is, co(t) = E[eltX] (-»<t<«0 = J " eitx dF(x) , where i = A/^T ¥e assume the moment generating function of X^ exists on a non-degenerate interval (a,b) and denote it by M ... Let K be the cumulant generating function. Then M(it) = cp(t) , / -~<t<» , and • • K(t) = log M(t) . We take the domains of M and K to be the subset of the complex plane given by [z : a <.Re(z) < b} The distribution and density functions of the stan dard normal distribution will be denoted by N and n , res-pectively. . ' Let P denote the probability distribution function n n of E X./(J?TG) , (n = 1,2,...) ; that is, i=l 1 where . P denotes the ,nr-f old .convolution • of P . . More generally, if: u ^ 0 , Fn and P. will denote the distribu-n tion functions of E (X. - u)/ (,/n a) and (X. - ^) , i=l 1 1 respectively, while M will denote the moment generating func tion of (Xi - u) . 1.2 The Edgeworth Approximation. THEOREM 1.2.1. If lim sup |co(s)| < 1 , and the th \s\-** r absolute moment of F exists, then F fx) = N(x) + n(x) Y n"*k+1 R. (x) + o(n'*r + 1) • k=3 (n-») (1.2.1) uniformly in x , for some polynomials R-^,. . . ,Rr , _e ach depending on u-^,. . .u , the first r moments of F , but  not on n , _or otherwise on P _or r PROOF. See Feller [7], p. 515. The series in (1.2.1) is known as the Edgeworth expansion of F . The construction of the polynomials, n Rk (k = l,...,r) , is described in Feller [7], p. 509. With 7. Its.first few. terms given explicitly, this expansion for Fn is Fn(wn = N(wn) - rT*rif N(5)(wn)] + n-1[^N^)(wn)+^x| N<6>(wn)] 3 r5I iV ^n; T 77 '"3*4 ^wn; ~"9J"'\5 ? xfi ^ 2 N(8)(W ) + 2100 ,2, „(10), v , 154 00 \4 ..(12 N "TOT X3X4 Nl ;(wn) + -121- *3 N }(wj] -(1.2.2), where wn = (x - nu)/(Vn a) , \n = ctn/an (an denoting the nth cumulant of F ), and denotes the ith derivative of N . In practice, it is not advisable to go beyond the second or third term of the series (1.2.2), as a well-known disadvantage of the Edgeworth expansion is that it then tends to give negative values for Fn(x) when x is small or values exceeding 1 when x is large. 1.3 The Cramer Approximation Cases occur where, in ?n(x) > x depends on n , as "when computing P[ X _< y] = Pn(«/n y/a) (with w = 0) . Then (1.2.1) will fail to hold. Since in this case both • Fn(x) and N(x) converge to 1 as n -» » , a more appro priate criterion of the accuracy of the approximation is the relative error. In particular, we would like.the relation 1 - Fn(x) (1.3.1) 1 - N(x). " to hold when both x and n tend to infinity. This relation is not true generally, since, for example, in the case of the symmetric binomial distribution, 1 - Fn(x) = 0 for all x > Jn . The limit in (1.3.1) does hold if x = o(n1//^) , this being a consequence of the following more general result. It is due to H. Cramer [4] and was generalized to variable components by Feller ([7], p. 524). Let the function V be defined by the equation •\z3 X(z) = K(h) - h K^^h) + -|z2 (1.3.2), where- h is obtained as a power series in z by inverting the series oz = V : (rri). hr"1 (1-3.3) r=2 = (h) Equation (1.3.2) implies X(z) =• X5/6 + (X4/24 - \|/8)z + (x^/120 - \j\h/12 + \j/8)z2 +...••.. (1.3.4) THEOREM. 1.3.1 (Cramer). Suppose there exists a 00 _ number hQ such that j e dF(x) exists for all — CO h e (-h0,h ) . Let x _be a real number, which may. depend  on n , such that x > 1 and x = o(n2~) as n - « Then v , 1 - Fn(x) « [1 - N(x)] exp [£- HJL-mi + 0(^)] m^y'* (n - .) (1.3.5a) For x < -1 , the corresponding relation is Fn(x) »N(x) exp [j^X (^)][l+0(^)] , (n - ' (1.3.5b) PROOF. See Feller [7], p. 517. When the Cramer approximation is applied, difficul ties may be encountered in the inversion of the series (1.3.3)* because the series given in (1.3.4) may fail to converge. Examples where.this occurs are given in section 3-3. The ap proximation can still be applied in these cases if it is pos sible to invert (1.3-3) algebraically. 10. 1.4 The Saddlepoint Method. We now present two saddlepoint approximations to Fn obtained by Rubin and Zidek [13]. They are based on an inversion formula derived in Lemma 1.4.1 from the Gurland [9] inversion formula. It asserts that -e T .. -ixt 1 - F ( ) = * + lim lim { f + f } Mn(it)dt (1.4.1) LEMMA 1. 4.1. Assume that the moment generating  function M exists in the region (z : a < Re(z) < b] . Then . p-e .T Y -ixt seio:uT+u^M(it)dt = W- I_l e-<c+iu>x MV+lu)!^ - t sign ( = ) • ' (1.4.2)" for every real number c ^ 0 such that a < c < b 3 where sign(t) -1 t < 0 I 0 t = 0 1 t > 0 V PROOF. • Suppose b > c > 0 , and consider the line integral in the. complex plane of the. function 11. f(z) = e"ZX Mn(z)/(27riz) along the contour I = I1 I2 +...+ Ig where, for fixed positive constants T and € (e < T) , U : Re(z) = 0 , e < Im(z) <_ T] , J2 = : Irn(z) = T , 0' < Re(z) _< c} , h = {z : Re(z) ---- c , -T < Im(z) < T} h = : Im(z) = -T , 0 <_ Re(z) < c} T5 = U : Re(z) = 0 , -T < Im(z) <_ - e] z6 = (z : z = €6 ie i 2 - 2J Then f f(z)dz = 0 . J I Now, . c • it - f(z) dz | f | exp (-x(y+iT)) Mn(y+iT) | dy JI2 ^ J o • |y+iT since ' • ' / ' • •/ [^(y+iT) |; = |E(exp[(y+iT)(X1 + . . . + Xn)])| < E(exp[y(X1 + ... + Xn)]) = ^(y) • Hence, If f(2)dz| <.1 t^-|(l - e"cx)] , 2 where A = max Mn(y) , and therefore o<y<c 12. lim f f(z) dz = 0 Similarly, lim j f(z)dz = 0 T— i4 Consider J f(z)dz . By the residue theorem, x6 i , v f(z)dz = 1 :(e) v where c(e) is a circle with centre at the origin, radius' e , and the integral is taken in a counterclockwise direc tion. ~ We shall now show that TT/2 . fl .. -77/2 . a lim f f(eeie)eie1B dfi = lim f ' f(eeie)eieip dfl :e-o\ J -TT/2 e-o TT/2 (1.^.3) Prom this it follows that lim f(z)dz = \ , and the desired result is an immediate consequence. Now, TT/2 f e-X€(cos 9 + i sin e> Mn(e cos fi+e i sin e)d -TT/2 1 f ' -xefcosP + i sin9)wn/ „ , . . -75= e v ;M (e cose + e 1 sine)de| W JTT/2 1 TT/2 = |W f [-e-xe(cosfi + 1 sin*)Mn(e cose +-e i sine) J-TT/2 13. _ e-x€(-cose + i sine)Mn(_£ cos0 + g ± sins)]d0| < 1_ JV2 |e-xe cos9 Mn(e cose + e i sin9)-eXe COse -TT/2 x Mn(-e cose + e i.sinp)|de By;the Legesgue bounded convergence theorem, the last quan tity converges to 0 as . e -• 0 and the result, (1.4.3), is established. Combining the results for J f(z)dz , f f(z)dz , and f f(z)dz , we obtain (1?4.2) for c > 0 . For a < c < 0 , the proof is similar. We shall make use of the inversion formula (1.4.2) in the form, 1 l'lC= 1 - FN(^ x/a) - -1(1 - sign(c)) + TJ^-.J exp[-nx(c+iu)+n logM(c+iu)] — 00 where we shall choose c to be a saddlepoint of the exponent in the integral; that is, c is the solution of the equation d [-nxz + n log M("z') ] = 0 (1.4.5), dz or t^1 - * t^-v-When c = 0 , the integral in (1.4.4) will be understood to mean that of the Gurland inversion formula. Daniels [5] showed that (1.4.6) has a single real root under fairly general conditions. 14. THEOREM 1.4.1 (Daniels). Assume that (t) = eK^ = f etx dP(x) M converges for real t _in -c^ < t < , where 0 < c-^ _< <» , 0 < Cg •<_ <° • Suppose F(x) = 0' , • x < a , 0 < F(x) < 1 , a < x < b , F(x) = 1 , b < x , where, possibly, a = -» o_r b=» _or both. Then (i) a and bo are finite if and only if K(t) exists for all real t , and (1.4.6) has no  real root whenever x i [a,b] , (ii) for every x € (a,b) , where -«<a<b<», there exists a unique simple root c of_ (1.4.6)., and K^?~\t) increases continuously  from x. = a to x = b , (iii) for every x e (a,b), where ?a and b may be infinite, there exists a corresponding c in (-c-^Cg). if lim K^^t) = b and (1) t-»+c2 lim Kv (t) = a (these conditions are satis-' t—c^ fied automatically unless a or b _is in finite). ' PROOF. See Daniels '[5]. Since K(t) converges for -c-^ < Re(t) < c2 , and c e (-c-^Cg) , K(t) has a power series expansion about t = c with a non-zero radius of convergence, and hence a. uniformly convergent series expansion for all t such that 11 - c | _< p for some p > 0 . Then, for values of n in some neighbour hood of the origin we can write 2 -nx(c+iu) + n K(c+iu) = -nxc + K(c)n - n u K(2) (c) "5 + n £ K^r) (c) (iu)r (1.4.7) r=3 rJ Let a* - [K<2> (C)]* , ' br = K<r> (c) ir/(rJa*r) ar = (-i)r/(ca*)r , (1.4.8) K(c,n,x) = J2T e"nxc + nK<c> , I(x,n) = ^ J" E-NX(C+IU) + N K(C+IU) du c+iu Then, proceeding formally, cm -n rlh K ; (c) (iu) -na u I(x,n) = K( c,n,x) J e" v r.' e 2~ ./2TT du -» c+ica ^Klc^x) j (l+Cff;^) e r=3 iy ,-1 n v br(y/,/rI)r _y2/2 dy ca ./27m (1.4.9) 16. Now, / __iy__v-i » 2 \1 + Ca*/jj ) = 1 +' I ar• (y//n) IyI < caVn , • • (1.4.10) and eo vr nib (y/./n) _^ • 6 e r=3 = 1 + n 2 b^ y^ + n 1 (b^y4 + # b^y ) + n"V2(b5y5 + b5b4y7 + £ b^y9) + n-2(.b6y6+[i-bf+b3b5]y8-^b2b4y104ifb4y12 + ... (1.4.11). Hence, _^ r=3 */n m=0 ,/n (1.4.12), where ; d0(y) = 1 • ' ! ' 2 4 2 6 d2(y) = a2y + (b4 + a-jb^y + | b^ y d^(y) = a4y4 + (b6 -Ka.^ + agb^ + a^b^y6 + (|b2 + b^ + a±b3bk + ta2b2)y8 + (^b2 + ^b^y10 + b^y12 M:SM>.\; .v:}K.. •;•'• , ; '•''•; (1.4.13), and, in general, d2k-l^y^ is 3X1 odd polynomial in y , while:•d2k(y) an even polynomial in y (k = 1,2,...) As odd powers of .y vanish upon integrating, the explicit form of •.'d2k_1 (y) (k = 1,2,— ) is not required below. It will be shown in the following chapter that • 17. 2m I(x,n) ~ K( j;*"'*) T, d V^=; (1.4.14) is an asymptotic expansion. Here, d2m = f n(y) d2m^) ^ (1.4,15) — CO 1 _ 2/2 (recall that n(y) = e ^ ) , and the series in (1.4.14) is obtained formally by interchanging summation and integration in the expression obtained from (1.4.9) by replacing the first two factors in its integrand by their series expansion. The first few coefficients in (1.4.14,) are d2 = a2 + 3(b4 + a-jb^) + b| (1.4.16) d4 = 3a^ + 15(bg + a-j^b^ + a2b^ + a^b^) + 105(tbi| + b^b^ + a1b3b4'+ |a2b|) +'945(t'b4b| + ^b^) + b^ . Explicitly, equations (1.4.16) are, with K^r^ (c) = K° (r ;=• 1,2,...) , do = 1 : d2 = -(ca*)"2 + a*-4'^ - K5) - -IT 4(o*r6 (1.4.17) , d4 = 5(ea*)-4 + | a*"6(- ^ Kg + ^ K5- ^ + ^ c c 35 *-8,l K2 1 1 „ 1 2v ,35 *-10 + "2T a ^T6K4 + ToK3K5 " "2^K3K4 + ^2K3} T6 a 18. , 12 1 3s 385 *-12_4 x (- ^1^+ ^K5) + 3^ a . Thus, d . x . Fn(>/fx_). ~ *(l-8lgn(c)) + e-™c+nK<c>(l + ^ + ^) 072701X2 n (1.4.18) , where c is obtained from (1.4.6) and dg and d^ from (1.4.17). Let us now return to (1.4.4) and by an alternate argument arrive at another saddlepoint approximation to Fn„ .. On letting p = c JnK^ and b ' = br/ir (r = 1,2,...) , we have, by regrouping the factors in the integrand of (1.4.4), CO r / rrr? (1.4.19) 1 " PnrbVr/= • *(l-sign(c)) + K(;c,n,x) f ' .n(u| e r=3 CO S;b ,{iu)r/;/Hr"2 ,v • r=3 - v: ". - » •-, But e = £ gT,(iu)(/-J , where Sr(y) (r'- 0,1,2,...) are defined in the obvious manner from equation (1.4.11). • • ' . ' / Define Qk(p) (k = 0,1,2,...) by Q(p)=J n{u)_ (iu)k du (1.4.20). -» p+iu Then, Vp> = nTpJ + fi1^0)! ' N(P)) (1.^.21), Qx(p) = 1 - p Q0(p) ; (1.4.22), 19. and . Qk(p) (k = 2,3,...) satisfy the recurrence formulae Q^p) = "P ^2k-l(p) (k = 1>2»'") (1.4.23), Q2k_1(p) = (-i)k"1(2k-3)(2k-5)...(3)(l)-pQ2k_2(p) . (1.4.24). i Thus, formally, we obtain, by interchanging summation and integration, a result of the form -oo k 1 " PnV^r) = ^(l-sign(c)) + K(c,n,x) I hk(p)(—i (1.4.25), n k=0 Vn where hQ(p) = 0.0(p) , h-^p) = ^ a*_3K3Q3(p) , h2(P) =^74 a*'4K4Q4(p) + a*"6K2Q6(p) , (1.4.26) h5(p) = a*_5K5Q5(p) + ^ a*-7K3K4Q7(p) 1 -*_Q "5 , . + i296 ° K3^(p) ' - Mp) = Y5o a*"6K6%(p) + (TT55K4 + T!Ok3K5)o*"8q8(p 1 *-10v2I_ A / \ , 1 *-12I_4_ , v + THZ8 0 K3K4Qloip) + 3110^ 0 K3Q12(p) , and where the , Qk(p) (k = 0,1,...,13) are readily obtained using equations (1.4.21), (1.4.22), (1.4.23) and (1.4.24). The approximation obtained from (1.4.25) by deleting all terms involving h^ (k _> 5) will be referred to as the saddlepoint 2 approximation. ! 20. 1.5 Remarks1 This last approximation will be shown, by numerical means in Chapter 3* to be superior for small samples to both the Edgeworth (1.2.1) and the Cramer (1.3.5) approximation, as well as to the saddlepoint 1 approximation (1.4.18), which fails for c = 0 . A family of formal series expansions of . 1 - P can be obtained in the manner of (1.4.25) by letting c take any value in (-c-^Cg) . In particular, c = 0 yields the Edgeworth series, which Cramer [3] proved to be asymptotic. Hence, if the solution c of (1.4.6) is 0 , the Edgeworth and saddlepoint 2 series must be identical. However, as c moves away from 0 , the quality of the Edgeworth approxi- •• • mation deteriorates. The explanation for this lies in the fact that c is a saddlepoint, and, as is well-known, through any saddlepoint there is a path of steepest descent (see, for example, [6]): As Daniels [5] has argued, the path of inte gration . [z ':. z =. c+iu,, < u .<<»} will closely approximate the path of ,'steepest, descent locally in' the' region near the saddlepoint, from which the 'Only' asymptotic contribution.,to the integral comes. 21. CHAPTER II THE SADDLEPOINT APPROXIMATIONS In this chapter we prove that the expansions given I in (1.4.18) and ^1.4.25) are asymptotic. To do so we use Laplace1s method for integrals and thereby achieve a more straight-forward proof than that provided by the method of .steepest descent (see, for example, Watson [.16]). Further more,, this method is readily adaptable to other applications • involving:parameters'different from, n which tend to infi nity.-•' • An example will be- given.' in Chapter 4. • 2.1 Asymptotic Expansions Returning to (1.4.8),, for c ^ 0 , I(x,n)=^fe° e-nx(c+iu> +nn K(c+iu)_du_ (2.1.1). -» c+iu iu -1 7 Let g(u) = (1 + —)" . Then g(u) is a integrable over the interval (-R,R) for all R and is equal to the convergent power series g(u) = 1 - AH + (IH)2 _ (IH)3 + ... . {2.1.2) in the interval [-6^,6 ] , where 6^ <c 22. As we shall show subsequently, the asymptotically dominant contribution to the integral in (2.1.1) comes from that portion of it which is over [-n ,n~^~^]. In order that g(u) be adequately represented over this region by th.4 1/3 first few terms of its series expansion, n must be large in comparison to c~^" . Since in the case of the saddle-point-1 approximation we assume g(u) is so represented, it tends to give unsatisfactory results when c is near 0 and ,n . is moderate1.,-,; The, saddlepoint 2 approximation leaves g . in its original form and tends to give good results even when / c ' is'small. ' However,, it is somewhat more complicated than the saddlepoint 1 approximation. ' : Let CO r h(u) = I a u , |u| < 6 , r=2 r h1(u) = -ixu + K(c+iu) - K(c) , (2.1.3) where ar = K^r^(c) ir/rJ (r = 2,3,...) and 6 is some positive constant. . / Daniels [5] showed that if the moment generating ' function M exists in a region D = [z : -c^ < Re(z) < c2] where D is the largest such region, one real root c of K^^(-z) - x = 0 exists, and it satisfies -c-^ < c < c2 As M(z) is analytic in D , M(z) = M(c) + M^l)(c)(z-c) + M^2)(c) (z-c)2 + ... (2.1.4) 21 23. converges in a circle with non-zero radius of convergence R Therefore, (2.1.4) converges uniformly on {z : |z-c| _< p} for some positive p < R . Hence, -log M(z) = K(z) has a power series expansion about z = c which converges uniformly on| [z ; | z-c | _< 621 for some &2 > 0 . Thus h-L(u) has a power series expansion which is uniformly convergent for i u j y>< &2 . \ that is, hx(u) = h(u) l2 = a2u + £ a ur,. ( |u| < 6 ) v (2.1.5). r=3 ' LEMMA 2, 1.1. Let 6 = min(6-L,62) . If 00 , I icp(t)ij' dt < (2.1.6(a)) for some real j > 1 ,• then -6 and r- - nh,(u) j g(u) e lV du =.0(n-M). , nh, (u) [ g(u) e 1 du = 0(n"M) 6 (2.1.6) for each positive integer M PROOF. If the [X±] ,. (i = 1,2,. . . ) , do not have a. lattice distribution, M(c-fiu) M(c) < p < 1 24. since |u| > 6 > 0 (Daniels [5]). Condition (2.1.6(a)) implies, 03 . after a change of variable, that J |M(c+it)|J dt < » for -oo ^ oo nh-.(u') some j _> 1 , say k . • Writing LN = ""TM- J g(u) e du n we obtain !LN! < -TM .fn"k:.|M(c+iu)|k du - A.nM/p-n for some constant A > 0 . Thus lim |L I = 0 , since n^» P < 1. . r"5 nni(u) M Similarly, J g(u) e du = 0(n ) LEMMA 2.1.2. There exists a positive constant p such that. Re{h(u)} _< -pu for all |u| _< 6^ where 6^ is  some, positive constant. PROOF. Explicitly, when |u| < 6 , h is given by .h(u) = S K^(c) (-ljr u2r + i E K<2r+1\c) >l)r u2r+1 r=l (2r)J r=l" (2r+l).' Therefore, / Re{h(u)} = -K2 + K^ u2 - Kg u" + . .. u where Ki=K^1^(c) (i'=2,3,...) . Now, Re{h(u)}/u2 is continuous, and equals -Kg at u = 0 . Using the ter minology and result (ii) of Theorem 1.4.1, K^^(t) increases continuously from x = a to x = b . Hence, Kv ;(c) > 0 . 25. Select € subject to the requirement Kg > e > 0 By the definition of continuity, there exists 6-^ > 0 such that if |u| _< 63 , * Re{h(u)}/u| _< -Kp + e < 0 ^ 2 Thus,.; • Re(h(u)} _< -pu for all |u| _< 6^ and for some P > 0 ' . (Prom now on, let ' 6 = min( 6^, 6g, 6^) . v ... \ First consider the saddlepoint 1 approximation. We expand. g(u) exp(n S a ur) = g(u) exp^nu^ Y a.^1"" ) " r=3 r=3 double power series in the two arguments nuy and u , convergent for all |u| < 6 . Denote this power series by • P(nu5,u) = E r c. . (nuY J (2.1.7) i=0 j=0 1J where the c. . (i = 0,1,... ; j = 0,1,...) are independent of n. and u ' . In order to approximate P uniformly, by its partial • "5 ' sums, we restrict nu^ to some finite interval, say, |nu5| _< 1 , that is, |u| _< n'1^ = 6n . . ' _3 We can assume n > 6 , so that &n < 6 . Then our region of integration for I(x,n) consists of five in tervals: (-»,-6) , (-6,-6n) , (-6n>&n) > (6n>6) and (6'ro) • If 6 could be allowed to depend on n , we could take 6 = 6n and thereby obtain only three sub-intervals of inte gration and a simplification of the proof. However, since 1 •26. 6n - 0 ,' |M(c+iu)/M(c)| would not then be uniformly boun ded by p < 1 for 6 <_ u < eo and all n , as it must if the result of Lemma 2.1.1 is to hold. LEMMA 2.1.3. .Using the same notation as above, ; j"6n g(u) e nh<u> du + f g(u) enh(u) du = 0(n"M) (2.1.8) - 6 c n for 'each positive integer M PROOF. For u e (6 ,*) , 2 .2 pn(u - 6n) > pn 6n (u - 5j > p(u - -6j ' (n > l) n n' n • Then, 2 2 2 2 pnfin ' r5 ~Pnu ' pn6n f™ -Pnu e j e du < e J e du 6n 6n • » -p(u-S') _i < e n du = p . . . n 6 -pnu2 y -pn1/5 . Hence, J e du = .O^e / J From this and Lemma 2.1. ^n f6 g(u) enh(u> du + J"5n g(u) enh^u) du 6n :- "s < J :|:g(u)||enReh(u)|du + f"n |g(u)||enReh(u)|du n -6. < f e"pnu du + f n e"pnU du 6n -6 27. = 0(e pn . ) (n > 6 •?) (.2.1.9) = 0(n-M) for any integer M > 0 and the conclusion follows. COROLLARY 2.1.3. For each integer M > 0 , f e-pnu.2 uM du = ,0^e-*Pnl/°; (2.1.10) 6n PROOF. If u _> 6n , we have uM e-|pnu2 < UM e-|pu2 < K (n > 1) for some constant K independent of n' . and so 2 uM = 0(e-nu) = 0(e^pnu ) . Equation (2.1.10) follows by replacing p .with ^p in 2 1/3 J e-pnu du = 0(e-pn ) It is now clear why c is chosen to be a saddle-point, that is, a root of equation (1.4.6). If this.were not the case, h(u) would include a linear term in U , and P would become E T, c'-- (nu) uJ . Restricting nu to 1=0 j=0 ^ a finite interval in order to approximate P uniformly by -1 -1/3 its partial sum leads to 6n = n rather than n The proof of Lemma 2.1.3 will fail since in this case 2 n 6^ 0 rather than » as n - co In the remaining interval, (""^n^n^ 3 P is l28. approximated by it's partial sums. For any positive integer A we write P (nu5,u) = r • c.. (mr3)1 uJ' . A i>0,j>0 1J i+j<A LEMMA 2.1.4. If |u| < 6n , P(nu3,u) - PA(nu3,u) = 0[(nu5)A+1] + 0[uA+1] (2.1.11), uniformly with respect to u and n (but not necessarily A), PROOF. Suppose an arbitrary power series, £ d zmyn , converges for |z| < R , |y| <S . , iri>0,n>0 11111 Since the terms of a convergent power series are bounded, dmn = °(R"ms"n) • Then, if |z| < R and |y| < S , E ^ dmn ZV = °( * i||m|||n) m_>0,n>_0 mn m>0,n>0 K & m+n>A m'+n>A / = o( r (||| + |||)k) k=A+l K b = 0((||| + |||)A+1) . Since, in general, |a +•b|r <. 2r_1(|a|r + |b|r) for all r > 1 , it follows that £ , dmn ^'V1 = 0( |z|A+1) + .0( |y|A+1) m>0,n>0 m+n>A 29. Equation (2.1.11) is an iriiraediate consequence. We now come to our main theorems in which we shall prove, with the aid of the preceding lemmas, that the ex pansions given in (1.4.18) and (1.4.25) are asymptotic. THEOREM 2.1.1. Let d. = (-a9)"H I c 0. m(-a9)m l ^ 2' m=Q m,2i-mv 2' xTfrn+i+i) > (i=0,1, 2,. . . ) , where (2) a2 ~ "K (c)/2 * the tcmn} are defined in equation (2.1.7), and p. denotes the- gamma function, that is, (2n).' . i 1 r(n+*) = nTi25""""" (2.1.12). Then,, if •, J M J < .» for some j _> 1 , CO • nhn (u) » . i ; j g(-u) e ' 1 -o Y. d. n"*"1 (n—) (2.1.13) • v-:.. , . . i=o a. is;an asymptotic expansion. PROOF. From (2.1.10) we obtain, recalling that a2 < 0 , / f,-8n. ' ,» i ^ na?u2 . !a2n1//5 {J. n + j. } PA(nu^,u)e ^ du = o(nAe d • j (n-«) -eo 6 n (2.1.14) for any fixed A Hence, combining the above and the.results of Lemmas 2.1.1, 2.1. 2, 2.1.3, and 2.1.4, :/•: « nh (u) f" napu i|J g(u)e 1 du " J_w PA(nu5,u)e 2 — 00 30. 2 dul •r^-'S nh^(u) , -6 .6 . nh,(u)' < I{J ^/}s(u)e du | .+ • |.{[ n+J }g(u)e ^Vdu| n 2 .-6 na„u + I If n + J } PA(nu3,u)e 2 du| 6 u2 + |J n[g(u)enh(u^ - PA(nu3,u)ena2U ]du| ~5n *a n1/3 - 0(n"M) + 0(e* 2 nA) 2 .oo na~u — CO + o(J. e 2 (|nu3|A+1 + |ulA+1)du) (n-~) (2.1.15), Now let us consider integrals of the type^ > r— -tx1- k e. x dx , where Re(t) > 0 ... For even k , sub-stituting'. tx = y ,•we obtain r e -tx2 x2m.dx = t -m-| r(m+J)-= ,-m-i!^^ (:m=0,l,...) mJ 2 (2.1.16). The folloxving estimate is valid for both odd and even k : 00 2 | |e-tx xkjdx = 0([Re(t)]-*(k4"1)) .(2.1.17) — oo using (2.1.17), we see that the last term in (2.1.15) is 31. 0(n"-A_1) . Therefore, combining (2.1.15), (2.1.16) and (2.1.17), •eo m>0,k_>0 'm+kCA x (.a )-*(3m+k+l) p(-|[5m+k+i]) + o(n-*A-1) + 0(n"M) (n-) , where is 0 or 1 depending on whether i is odd or even, respectively. As A and M are completely arbitrary, ! we obtain an asymptotic series » nh.,(u) » i . g(u)e du ~ E d.n * (n-») . -,M -co . i=0 1 where the -d^ are. the coefficients computed in Chapter 1 (1.4.17) and given, in the statement of the theorem. J''; „•:•'*•' •'- • A, \ •'' nu^ E a u - • '.- .•' ••" . r=3 r » „j K(j) . • THEOREM 2.1.2. Let e = E • -if E j=0 J' k=3j x c, ({a })uk , where a = K(r)(c)ir (r=3,4,...) , ri K(0) = 0 , K(j) •'= 00 (j > 0) /'and - ck({ar}) (k=0,3,4,. .. ) are appropriately defined constants depending only on . (ar) (r=3,4,...) . Then, with \(z) (k=0,l,... ) defined  as before (see equation (1.4.20)), g(u)e ^ du ~ E E ' c \) .C/2T -» j=0 k=3J (_2a2)r/2 x nj_k/2 Qk(c,/C2lva^)/j.' ' (n—) (2.1.17) '32. ••is-, an^asymptotic expansion. PROOF. The proof is almost identical to the proof of Theorem 2.1.1 and will only be outlined briefly here. . ' . nu5 2, a ur~3 In this case, the factor e * . in the integrand of (1.4.19) is expanded as a power.series in the 3 3 argument (nu^) . This series is denoted by P'(mr) As in the proof of Theorem 2.1.1, mr is restricted to the •interval |hu3| _< 1 , that is, |u| _< rT1^ = -6 , in order to approximate P' uniformly by its partial sums P^ Lemma 2.1.3 can be applied without alteration, while Corollary 2 2 2.1.3 is obviously valid with e"pnu• replacing e"pnu , c+iu since the latter.is less in modulus than the former everywhere. In the same manner as we obtained (2.1.11), we find that, for lul < 6 , n (P' - P')(nu3) = 0([nu3]A+1) A uniformly in u and n . Then / 2 \ [, g(u)e 1 du = I p;(nu3,u)e 2 -oo -a, C+XU + 0(n-«) + 0{f e^lnu'l^du} , (2.1.18), Writing p;(nu3) = E ( E a ur)J' j=0 J' r=3 33. 'A j K(j) k n = ^ ii_ £ c, ({a }) u , equation (2.1.18). becomes 3=0 J' - k=3j- K; • , c nh (u) A K(j) ,f ,-r nJ'_k/2 Qk(c,/T2naJ) j2lT/y. + 0(n_M) + 0(n"^A"1) (n-eo) (2.1.19). Again, as M and A are entirely arbitrary, the above, and hence (1.4.25), is an asymptotic series. This completes the proof of the asymptotic nature of the saddlepoint 1 and 2 approximations. We now discuss briefly a case not included in Theorems 2.1.1 and 2.1.2, 2.2 The Lattice Case. When the [X^ , (i=l,2,...) , have a lattice distribution, the preceding argument fails; the tails of the integral (2.1.6) cannot be ignored, since for a distribution having its mass concentrated at points h units apart, the characteristic function is periodic of period X = 2Tr/h , with |cp(X)| =1 and |c?(s)| <,1 for 0 < s < X . Daniels [5], in his work involving the density func tion, avoids this complication because he is dealing with densities instead of distributions. In that case, the path of integration stops at c + ITT , and no tail regions are present. Using this1 fact, an approximation to the distribu tion function could be obtained by numerically integrating the density function. Feller [7] introduces the concept of a polygonal approximant F* to Fn , where, more generally, G# is the . convolution of G v/ith the uniform distribution on (-h/2, h/2) and- h is the span of G . He then shows that the first two terms of (1.2.2) approximate with an error of magnitude> o(n"^) . This means that at the lattice points of Fn , the error is o(n~2) when Fn(x) is replaced by |-[Fn(x) + F (x-) ] . However, for higher order expansions of the type (1.2.2) the additonal assumption that lim sup |cp(s) | < 1 (2.2.1) |s|-« is necessary, a condition not met by lattice distributions and a considerable number of other distributions which have their variation concentrated in a set of Lebesgue measure zero. The order of magnitude of the error in approximating F by a series of the Edgeworth type depends on the arithmetical nature of the set of possible values of the random variable X. Even if all the moments of F are finite, in the case of ! discrete distributions it is necessary to supplement the expansion (1.2.1) with discontinuous terms. ' However, although it is impossible to approximate such"discrete distribution.functions with continuous functions to an accuracy of within .one-half of their maximum jump, local limit-theorems for approximating Fn at its points of dis continuity exist (see Gnedenko-Kolmogorov [8]). We reproduce the' following proposition of Esseen (see [8], p. 241) which is analogous to the Edgeworth expansion (1.2.1) in the absolutely continuous case. 35. Suppose the random variables X^ (1=1,2,...) can only take on the values x = a + sh (s = 0, +1, +2,...) , o """" where h is the maximum span of the distribution P .. The random variable •n n oM k=l K can only take on the values yns = h(S " nP)/(°^H) > where p = r, ipi and p^ = P(Xk = a+ih) . Let l=-eo Pn(s) = P(Yn = y^) . THEOREM 2.2.1. (Esseen). If the identically dis tributed random variables '. -X^,...,X are independent and have finite-, absolute moments of the order k(k >_• 3), then. ; ' Pn<s) = TJ^^ns^ r^.^±/2 \(n(Yns)) . + 0(n"(k_l)/2) '• (n—) . Here, Rl(n(yns)} = " 37 *0)(y) > D , , u Xk (k),• . 10 ,2 (6), x R2(n(y)) = TI n (y) + sr x3 n (y) ' and the R^(n(y)) (i > 2) are obtained in a similar manner from the expansion (1.2.2) by replacing N^^(y) by n^^(y) 36-PROOF. See Gnedenko-Kolmogorov [8], p. 241. To obtain the values of the function Fn at the points of discontinuity y now only requires a summation procedure, F (y ) = F (y , + 0) nuns; nVJn,s-l ; = S P (r) r<s / 37. CHAPTER 3 y. ' •;• .'. . "' COMPUTATIONS . . : ;,. ^To judge the quality . ofthe saddlepoint'approximations in the case, of small • n several test cases were considered. Numerical results were obtained in each case for the sake of comparison with the Edgeworth and Cramer approximations. These results were obtained for values of the argument, x , selected to represent the entire admissable range of values and for values of n between 1 and 40 inclusive. For bre vity, only a few representative results for each distribution considered are depicted. * . It will be noted that whereas the saddlepoint 2 expansion gives uniformly better results than the other three approximations, the Edgeworth series, (1.2.2), is quite good when c is close to 0 , as we would expect on the basis of the discussion in section 1.5. However, when x assumes values in the extremities of its range, the saddlepoint method gives substantially better results. 3-1 Remarks on the Tables. When e = 0 , the saddlepoint 1 expansion does not exist, and for programming purposes, the- Edgeworth ap proximation is printed in its place. When F (x). is nearly 1, exponents in the calcula tion of the Cramer formula are excessively large for the com puter, and the value F (x) =1 is assumed. 38. Multiple entries under the headings represent suc cessive approximations obtained by adding, at each stage, one more term to the approximations. They are included to faci litate a comparison of the rates of apparent convergence of the various series. The results are printed in exponential format, and a series of digits, say O.n-^. ..n^ , followed by "D +. m" +m represents the number O.n-^. ..n^ x 10— , The letter E occasionally replaces the letter D in this format. In order to observe the effect of the location of the saddlepoint on the various approximations, the value of c and the accuracy to which it is calculated is given. Hence, "saddlepoint = + or - 5" means c e (c^-6,c^+fi) When it is available from existing tables, the correct value of Pn(x) lS given for comparison. Prom these cases it appears that the last entry for the saddle- j • point 2 approximation in each case is accurate at least in the digits where it. and the next to last entry agree. In the .remaining cases,, judgement on the quality of the various approximations must be withheld7until exact values become /available. If .the last saddlepoint 2 entry is accurate to the extent just.described, as seems likely to be the case, , - an examination of the tables indicates that this method of approximation gives results of the same comparatively good quality as it did in the earlier cases for which exact values of F (x) are known. 39. 3. 2 Chi Random Variables. Let X± = |Yil where Y± (i = 1,2,...) are independent, standard normal variables. The density function of is given by ET 2/P ! f (x) = VTT e"x 7 , x 2 0 0 , x < 0 , and the moment generating function is M(t) = 26* /2 N(t) , 1 v2/2 where ' N(t) = J e y / dy — eo The cumulants are given by Rubin and Zidek [13] as ^1 = ai Z 0-79788 45608 03 a2 = a2 = 1-a2 ~ 0.36338 02276 32 a3 a1(a2-a2) ~ 0.21801 36l4l 45 a4 = 2a2(2-3a2) ~ 0.11477 06820 54 (3-20a2+24a^) ~ - 0.00443 76884 6262 The saddlepoint c is the root of the equation n(z )/N(z.) + z ='x/n which/can be solved numerically using Newton's Method. 40. In order to compute the correct value of F (x) , equation (1.4.4) was inverted numerically in [13]; This entailed the evaluation of M(z), and hence, of N(z) , for complex values of its argument. Since use of its Taylor ex pansion results in uncontrollable round-off error, an ac- • curate method using continued fractions was employed. The computation of . QQ(p) (see equation (1.4.21)) also requires W(p) , and for this reason, a detailed account of the me thod devised in [13] is given in Appendix A. The derivatives of the cumulant generating function evaluated at c are K-^ = x/n = x _ .. _2 K2 = ex + 1 - x K^ = c2x + c(l-3x2) - x(l-2x2) K4 = c\ + c2(l-7x2) - cx(5-12x2)' + x2(4-6x2) K5 = c4x+c3(l-153c2) - c2x(l6-50x2) - c(3-35x2+6dx4) + x(3-20x2 + 243c4) K6 = c^x + c^(l-31x2) c%(42-l80x2) - c2(13-191x2+3903ci|')'+ cx(4l-270x2+360xl1) - x2(28-120x2+120>c4) . n From now on, F (x) will denote P( £ X. < x) . Then, for n f i=l 1 ~~ the saddlepoint 2 method, the first term approximation to Table 1(a) 11 CHI RANDOM VARIABLES (=ABS'( V) .WHERE Y = NORMAL ? ME AN = 0 t VAR . = 1 ) N = IO X = 3.6 00 00 SADDLEPOINT = -0 . 2 1192 38'01 + 0R- 0.366E-13 F( X ) = 0.410254 E-02 EDGEV/ORTH CRAMER SADDLEPOINT 1 SADDLEPOINT 2 0. 0. 0 . 1080667090-4409401340-405517680D-01 02 •02 0 .4046812430-02 0 .4452880830-02 0.3978365120-02 0 . 4 174711670-0 2 0.38334 64480 0.^106579370 0.409961609D -02 -0 2 -02 0. 404377742D- 02 0.4 102644 590 0.4102950790 -02 -02 X SADOL EPOINT 7.00000 = -0.2939604 ,'E 00 + 0R- 0.5 55E-16 ! F( X) = -.. ; EDGEWORTH CRAMER ....SADDLEPOINT 1 SADDLEPOINT 2 0. 0. 3 0 33036570 31730 92 810;. 00 •00 " 0 .3014527160 00- 0.6766890570 00 -0. 1.67467490D 01 0.3015973490 0.318787413D 00 00 o. o: 3 18:6 01 1.58 0'' 3.1 88 1'48570 . V•'. X f oo. • 00 = 10.40000. • 0.2359028000 0'2 •• 0.31871.34320 0.3183.5 3 2480 .0. 3 18.353325D 00 00 CO . .' SADDLEPOINT F ( X ) = 0.5623426;'E 00. +0R- 0.6I9E-14 = 0.89276430 E 00 EOGEWOR. IH CRAMER SADDLEPOINT 1 • SADDLEPOINT 2 0. 8979775370 00 0 .8883722670 00 0.8 45 7 06 7.380. 00 0.83803012 30 00 0. 0. 0. 8922496760 8929280630 . 8 92 7336.62D_ 00 00 00. 0.9589.380730 00 0.7223566340 00 0.8 9250332 ID. 0.892749136D 0.8 92758476D 00 00 00 0.8927634590 00 X •= 13.90000 SADDLEPOINT = O.U 5635l',E OL +QR- 0.222E-15 .. F.(X.) _=... 0.99729399 E. 00 . EDGEWORTH 0.999052373D 00 0.9975984990 00 0.9 9 7.199 8 5.10 . 00 0.9973176900 00 CRAMER 0.996790.3210 00 SADDLEPOINT 1 0.9 9 696 46 2 50 0 0 0.997404934D 00 0.997230346.0 00 SADDLEPOINT 2 0.9972253000 00 0.9 9728188 50 00 0.9972943640 00 0.9972937810 00 0.9972939980 00 A2. i CHI RANDOM VARIABLES {= A8S(Y) »WHERE Y = NORMAL,MEAN = 0» V AR . = 1 ) N •= 40 • X =*""l8. 7 59 99 SADDLEPOINT = - 0. 1 3 079 50' :E 01 +0R- 0.1UE-14 F( X) •= 0.6274 E-04 . EDGEWORTH ' CRAMER . SADDLEPOINT 1. SADDLEPOINT 2 0.2796856250-03 -0.1671882140-04 0.5 64 75 6959 0-04 0.63725764.30-04 0 ."6323368520-04' 0.6 5 9743 3 76D-04 0.621543542D-04 0.6292393050-04 0.6128094370-04 0.628270354D-04 0 .6 2743 I9 54D-04 0.627446602D-04 0.6274432780-04 .' X; ';. . •'. = 24.75999 . '';S AODL E PC I NT •= -0 . 5843487: '.E 00 +QR- 0.694E-15 F(X) - = 0. 2541083 E-01 ., i ;:EDGEypriTfj.._y .CRAMER SADDLEPOINT 1 SADDLEPOINT 2 0 .302715069D- 01' 0.2465491280-01 0.298779195D-01 • 0. -2472592890-01 0 .257362.3 94 0-01 0.2 27366469D-01 0 .2542728.390-01 0 .254511868D- 01 0.2 818401640-01 0 .2 54 1.0 1692.D-•01 0 .254154082D- 01 0.254113 0 69D-01 -- . .. --. -• 0.2541123900-•01 X = 39.00000 SADDLEPOINT = 0.42 83156- E 00 +0R-. 0.416E-16 F { X ) = 0.96412521 E 00 EDGEWORTH CRAMER SADDLEPOINT 1 SADDLEPOINT 2 0 .9684337340 00 0.963448929D 00 0.9556373200 00 0.9633715030 00 0 .963 367 5460 00 0.,.9702510900 00 0.9640933810 00 0 .964158 50 ID 00 .•' 0.95638 55400 00 0.96V124178D 00 0 •.3641.2J.600D_ 00 0.9641242910 00 0.9641244540 00 X 45.00000 SADDLEPOINT = . 0.72 39 73 4; '.E 00 +0R- O.OOOE 00 F( X) = 0.999317.10. E 00 EDGEWORTH CRAMER SADDLEPOINT 1; SADDLEPOINT 2 0 .9997004530 oo • 0.999297909D 00 .0.9992547.280 00 0 .999 307 3420 00 0 .99938S199D 00 0.9993332-770 00 0.9 99.3 161.800 00 0. .9 9931.1 19.40 . 00 ..0.9.993 100 14D 00 0.999317116D 00 0 .9993169090 00 0.9993171000 00 0.999317104D 00 43-Fn(x) is . , >> Fn(x) sign(c)) - e-xc+nK(c) QQ(p)A/^ , where QQ( p) is given by (1.4.21). •"'•1V. The results xve obtained in this case are' listed in Tables 1(a), (b)." The exact values of Fn(x) 'given are those computed by Rubin and Zidek [13]. 3-3 The Exponential Probability Law. The probability density function of the exponential distribution is given by , x > 0 ,• otherwise where X > 0 . The moment generating function is M(t) = x/(X-t) (Re[t) < X) , from which we obtain the cumulants a± = K(X\O) = i.'/X1 (i = 1,2,...) . Solving the saddlepoint equation yields f(x) = Xe 0 -Xx c = X - (x/n) 44. and hence, K^r^(c) = r< X (x/n)r (r = 1,2,...) . Difficulties are encountered in applying the Cramer approximation. For certain choices of the parameter X and the argument x , series (1.3-4) does not converge. For example, when X = 1" , (1.3-4) becomes ' w ^ 1 1 . 12 1 3 „, X(z) = j - ^Z + ^Z- - -g Z + ... , which does not converge for |z| > 1 . Thus, if x > 20 and "" n =10 , vj/Jn = (x-nfi )/(no) > 1 , and x(w/7n) cannot be' evaluated using (1.3-4). We overcame this difficulty by inverting directly the equation (1.3-3). The form of the expression for c indicates that it can be considerably different from 0 for moderate values of x . Hence, the Edgeworth series often yielded inaccurate results. For example, when x = 4 and n = 15 it is incor rect in the first .-significant figure. Results for this case are listed in Tables 11(a), (b). , i 4_5 EXPONENT IAL RANDOM VARIABLE ' M E A N = 1 VARIANGE=1 15 SADDLEPOINT = 31 .OOOOO 0.5161290' .E 00 + 0R- O.COOE 00 F(X ) = 0. 99948'^ EDGEWORTH CRAMER SADDLEPOINT 1' SAODLEPG FNT 2 0 .99993 19550 00 0 .9972110870 00 0.999417655D 00 0.9994470520 00 0 .9998733900 00 0.9994914080 00 0.9 9 947 57 2 00 00 0 .999634452D 00 0.9994697970 00 0 .9994762350 00 c .9994320490 00 0.9994763730 00 0 .9994370310 00 0 .99947.63290 00 X 11.00000 SAOOL EPOINT '= -0.3o36363 E CO +0R- O.OOOE 00 F (X ) ••= 0.14596 .EOGEWORTH .CRAMER SADDLEPOINT 1 •SADDLEPOINT 2 . . 0 . 1 508490580 00 0 .1339353650 00 0.2011847340 00 0. 1339272020 00 0 .149 50 70 06 0 CO 0,6 1752 53 640-0.1 0. 1454332 790 00 0 . 1466.16392D CO 0.3823763470 00 0 .-145 89 55 560 00 0 . 1460833740 00 0. 1459526560 00 0 . 145980.7790 00 0.1459570810 00 X 5.75000 • SADDLEPOINT •= -0. 1603695- E 01 +0R- O.OOOE 00 F(X) - 0.93 • E-03 EDGEWORTH CRAMER SADDLEPOINT 1 SADDLEPOINT 2 0.8462344 170-02 0.9447172850-03 0 .98556825 10-03 0.86498 30210-03 -0.86121708.20-03 -0.31842 6 7.690-03 0 .'59.6'495454D-03 C. 90 14119670-03 0 .913860382D-03 0.9347567530-03 0.^261830 780-0.3 0^9279023940-03 0.928.3587020-03 0.9284334050-03 X SADDLEPOINT F {X ) EDGEWORTH . 4.00000 -0.2749999.£ 01 0.2 '" E-04 +0R- O.OOOE 00 CRAMER SADDLEPOINT 1 SADDLEPOINT 2 0 .2254348950-02 -0 .2044016900-02 -0 .3932762050-03 0.2349010740-03 0 ,2 12706214D-03 . 2980789600-04 0.2062211930-04 0. 1982582900-04 0. 199 59 51.8 30-04 0. 1368297240-04' 0, 1990022450-04 0. 1992148070-04 0 , 1993001850-04 0. 1993159130-04 iab_le_jni(.b_^ EXPONENTIAL RANDOM VARIABLE ME AN = 1 VARI /-f,CE=l N = 4 0 X = 15.50000 SADDLEPOINT •= -0. 1530644 ;E 01 +0R- O.OOOE 00 F(X ) = EDGEWORTH CRAMER SADDLEPOINT 1 SADDLEPOINT 2 0 .5357784620-04 0 .179693256D-06 0 . 1527063360-06 0.1440901120 -06 -0 .108795561D-03 i 0 . 1484449270-06 0.14886^4730 -06 0 .6 180169920-04 0 . 148 9199 000-06 0. 1488 38 31.70 -06 ' 0 .4954126 660-05 0 . 148847695D -06 -0 .6977151760-05. 0.1488491500 -06 X = 30.00000 SADDLEPOIN i T" •= -0. 3333333 E 00 +0R- O.OOOE 00 F (X) = . EDGEWORTH CRAMER SADDLEPOINT 1 . SADDLEPOINT 2 0 .5692336810-01 0 .4401459090-01 0.5589016810-01 0 .4401024010 -01 0 .478 87 2 484 0-01 0 . 3900 667 99D-01 0.4619276910 -01 0 .4638122850-01 0.558087329D-01 0 .4624715080 -0.1 0 .4625760940-01 * 0.462 52 79430 -01 __p. .4 62.51 71560.-01.. - - - 0 .4625321250 -01 X 45.00000 SADDLEPOINT = 0.1111111 E 00 +0R- O.OOOE 00 FIX) = EDGEWORTH CRAMER SADDLEPOINT 1 SADDLEPOINT 2 0 .7854023250 00 0 .780228869D 00 0.621907080D 00 0.7802289270 00 0 .7911709600 00 0.1303262030 01 . 0 .7918598.340 00 0 .7916667020 00 -0 .2237106370 01 0 .7916026090 00 0 .79 16126760 00 ' 0.7916187610 00 0 .7916188290 00 0 .7916182510 00 :"' '• 'X ; - 55.00000 , '• SADDLEPOINT- ^ 0 „ 2727272. E 00 '7+0R- O.OOOE 00 - - ' F (X ) . - - . -::E,DGE WORTH . ^CRAMER;. . SAD.CLEPOINT 1. SADDLEPOINT 2 0 .991. 1469680 .0.0 0 .9846229.370.: GO,. •0.9824797820 00 A. 9 8 46454 8 70 00 0 .9 853069C8D 00 0.986799003D 00 0.9853127.530 00 0 .9.8 5 1330 3SD 00 0.9 839343960. 00 C.Q35301129D 00 0 .985341275D 00 0.9853029960 00 0 .98 5 29 7 5 50D 00 0.9353028100 00 rr ro r>~ 0" r/j; o o o o o o o o o o c O O o o o o o o O o a o o o c-o CJl o 1 1 1 1 I rv i 1 1 1 1 rv rv c C: Cc c r-, c c c c c o, c c CJ o o h-cr- CJ CJ c c- h- o--CT- cr- O'- \- r—• r~l i—i \— -.0 sC r- r- r- r- ~ZL > .—r o IP; LP; LP- LP LP O' o CJ c O 1— ro rp. rc'i rO rp> xT •4- r— G o o o o c <?• cr CJ cr- CJ c? r—t ^-i • O rv P.! (V: cv Ol C rv P0 rv p.; P>! eg f\J rv cv rs; c o o O cr o CL. vO ~r- a. C- CJ a- CJ CJ LL' o <; O - LL- 'J-- l.p LO LP-. LP . 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CJ- o-c--< • a • O <l « • <c « • < 9 9 9 GO . o o CJ O 1 UC- o o o ! o o o o o o CJ) CJ-o o w cc> 1 o LU w LU UJ H UJ vO ro H vO c> LT\ -cn O 'vO rA o :=r o c> O o o -c a> 1 C: <. r-i I o CM o OJ o OJ (_• o o c c- vO c CJ c- Co o CJ,' *ovo o LO LP. o •-C ^f- o o H c. •£ OJ Cd . <r C" 0- G- o H Cr MC P; <7' r<\ o: o ~ ro\ Lb o 0J IT. c- LiJ o 0* LL; o 0 --0J cj « OJ • .—t X- rv 9 OA, '.'i rv (Xi • < o 0 <' • <. CJ 1 O O DC • xr> 1 o o cc LO. c 6 .' Q: NT' d rH 1 ' rvi 1 CJ r-i p P! r-0 IV C ro H II II r\i II II II rvi n II II II II n * j « * 1— 1- a 1~ i cj. h- o z. J-_ V—- •\ 1—* o 1 c j a c : ro co P~ ro ro ,—1 i—1 r-i a. o o o Q- o o CJ- o o LU O O o O ai o o o o. LU C O c o o LL.I o o o o o CJ " 1 : c, 1 c 1 c. 1 a I c , E , ,. ; 1 c 1 o 1 . 1 c, 1 • Q .— c a o E —. c o c c-c x x o c o - c- X T; LTi u". ir, C: X X o o o o o C; X ZL.' -i <. — i •4- <r~ < ~— H LH IP. u > LP; LT'i — •ii <l" — H-X ii. ex. o CJ c-o X a. IP, LP. LP-. LP LPl X L/) U- • ex. o- ST X ij"i CC <J u.. -o CJ . cr CJ- CJ CJ CJ ; a r~l .—• .—t r-^ --i • ; O rv pj rv P4 PJ C: rv cv CV cv cv PJ P-J rv P-J rv ' Jr-" o o o C~' o . Jf- >£ o .J- CJ Ch Cr- CJ-LL' •ij .^J ^0 ; Llj LP LO u-i : U.: NT LiJ r—1 r~1 —' " ;JJ rv rv p. cv rv Ci! r- r- P- r O r— r—( r-1 r-i .—i r<i CO ro m O CO CP ro ro ro ! C CV rv PJ rv oj • o o CJ <Ji V G ro ro r<~. CO i-p. UJ rv rv rv c-J ; LJ rv r-vi rv rv CV! ' UJ --o LL; CJN CJ- 0S CJ-• • • a - « « e 0 9 0 * 9 4 • • • \ O o C o o o o o O i o c o o i o o o O o o .Table III CM, 48. NORMAL RANDOM VARIABLES • MEAN- .5 VARIANCE^! N = 3 6 SADDLEPOINT F (X ) ' 0.00000 • -0. 5000000''F 00 +0R- O.OOOE 00 Q.1349898'03 E-02  EDGEWORTH CRAMER SADDLEPOINT 1 SADDLEPOINT 2 0 . 1 3 4 c'3 97 75 D-02 0.13493 9775 0-02 0 .1349897750-02 0.1349397750-02 0.1477282800-02 0.1313140270-02 0.13678 54450-02 0.13493 93 030-02 0.1349B98030-0? 0 ; 1349898O3D-02 0.1349897750-02 0.134989775 0-02 0.1349898030-02 0.13498 98030-02 X = 9.0 0000 SADDLEPOINT = -0.2500000 E 00 + 0.R- O.OOOE 00 F ( X ) •= .0.668072013 E-01 . EDGEWPR.TH/. CRAMER '_. .. SADDLEPOINT 1 S A DDL EPOINT 2 . . . 0 .6680733650-•01 0 .6630733650-01 0 . 8634506380-01 0.6680720130 -01 0 .6 68 07 33 65 0-•01 0 .4 79 69 47 990-01 0 .66807201 3D -01 0 .6680733650-•01 0.9913692510-01 0 .66807 20130 -01 0 .6680733650-•01 0.6680720130 -0.1 0 .668.07 33.6 5 0-•01 - •- : • - .0 .6680720130 - 0 1 X 15.00000 SADDLEPOINT •.— -0.8333331 E-01" +OR-;:B'-.000E 00 F ( X J ZZ 0. 308537539 EDGEWORTH CRAMER SADDLEPOINT 1 SADDLEPOINT 2 0 .3085375400 00 0 . 30 35375400 00 0.704 1.306540 00 0 .30? 5375390 00 0 .3085375400 00 -0/. 21 12 39 1 960 01 0.3085375390 00 0 .3085375400 00 0.316858794D 02 0 . 3085375390 00 0 . 3085375400... PC .... 0. 303 53 75390 00 0 . 303 53 7540D 00 0 . 3085375390 00 X = 24.00000 SADDLEPOINT = 0. 1.666666'-E 00 '+0R- O.OOOE 00 F(x) . • = 0.841344746 EDGEWORTH' CRAMER 0.8413447370 00 0.8 41344 7 370 00 0.841344737D 00 0.8413447370 00 0.84 1 34.4 7 370 00 0.8*13447370 00 SADDLEPOINT 1 0.7580292750 ..00 0. 1 000000000 '' 01 .0. 2 7403 78 260 .00 SADDLEPOINT 2 0.841 3.447460 00 0.3413447460 00 .0 . 841 3447460 00 0.3413447460 00 0.8413447460 00 49-3.4 The Normal Probability Law. : This case is considered because extensive and . highly accurate tables are available. The results obtained indicate the high accuracy possible with the saddlepoint 2 approximation. In all cases considered, the first term in (1.4.25) gave an answer which is correct to every figure tabulated. However, as the results given in Tables Ill(a) and (b) indicate,•' the Edgeworth and Cramer methods generally incorrect in the last two or three figures. 3.5 The Non-Central ChirSquare Probability Law. The distribution function of the [Xi) (i = 1,2,...) is given by F(x|v,\) = T. e~;/2 (X/2)J Fjx,y + 2j) , j=0 j.< C where X _> 0 is termed the non-centrality parameter, v is,the number of degrees of freedom, and F fx,k) = [2|k P (ik)]'1 fX t|k_1 e-^ dt (0 < x < .) u o .' ~ is the central chi-square distribution v/ith n degrees of freedom. The characteristic function of X^ is cp(t) = exp[Xit/(l-2it) ](l-2it)_V//2 Using equation (1.4.6), we readily find that c, the saddlepoint, is given by 50. •- 2 Also, the derivatives of the cumulant generating function K evaluated at t = c are given by K^\(c)-= (l-2c)"J' 2J'-1(j-l)j [v + \J7(1-2C)] '. As a first approximation to Fn(x) we can write, using (1.4.25), Fn(*) - + sign(c)) + e-xc+nK(c)+p /2[|(l + sign(c)) - N(p)] , where p - cvnK^2^(c) . Numerical results are tabulated in Tables IV(a), (b) and (c). Table •IV(a) • ^ NON-CENTRAL CHI-SQUARE 1 DEGREE OF FREEDOM NON CENTRALITY PARAMETER = 2 X = . 0.5O00O ' SADDLEPOINT = -0.5354101 E.01. +0R- O.OOOE 00 F { X) = SADDLEPOINT 1 SADDLEPOINT 2 EDGEWORTH 0.2015249080-01 -0.4626224490-03 -0.4282432470-02 -0.350359347D-02 -0.236738097D-02 CRAMER 0.408479631D-03 0.7625912710-04 0.744184606D-04 0.7458564880-04 0.6115440530-04 0.7279859780-04 0 .7379074130-04 0.7431073410-04 0.7448304920-04 X = 10.00000 SADDLEPOINT = -0.1403881 E 00 +0R- O.OOOE 00 F { X ) EDGEWORTH 0.239750076D 00 0.260255003D GO CRAMER 0.225509436D 00 SADDLEPOINT 1 0.4153531 190 00 -0.1803945270 00 SADDLEPOINT 2 0.226020105D 00 0.258417464D 00 0.260098775D CO 0.2602656350 00' C.2602844530 CQ 0.2696189500 01 0.2596917290 00 0.2602251960 00 0.2602753060 00 20.00000 SADDLEPOINT = 0.7846480 E-01 +0R- O.OOOE 00 F (X ) EDGEWORTH CRAMER SADDLEPOINT 1 SADDLEPOINT 2 0.7602499240 00 0.7508088640 00 0.5424134760 00 0.7506725330 00 0.7807548510 00 0.780911079D 00 0.7810779390 00 0.7810591210 00 0.1713239310 01 -0.657541799D 01 0.781703890D 00 0.7807042540 00 0.7810869350 00 0.7810573610 00 X .= 50.00000 SADDLEPOINT = 0.2500000 E GO +0R- O.OOOE 00 F (X ) = ORTH CRAMER SADDLEPOINT I SADDLEPOINT 2 EDGEW 0.9999996280 00 0.999993707D 00 0.9999559180 00 0.999838539D 00 0..9996826.190 00 0.100000000D 01 0.999736361D 00 0.999769215D 00 0.9997609390 00 0.9997472650 00 0.9997612530 00 0.9997632710 00 0.9997632280 00 0.999763223D 00 52. NON-CENTRAL CHI-SQUARE 1 DEGREE OF FREEDOM NON CENTRAL ITY PARAMETER =2 N = 15 X = .. ' 0. 50000 . SADDLEPOINT = -0.1544097 E 02 +0R- O.OOOE 00 F ( X ) ;= EDGEWORTH 0.1398500810-03 -0.3643638920-03 0.2044586970-03 CRAMER 0.2933935700-09 SADDLEPOINT 1 0.8276460610-15 0,820674756D-15 0.8205665220-.15 SADDLEPOINT 2 0.7486926930-15 0.8176701410-15 0.819280304D-15 0.86.3652947D-04 -0.1594743060-04 0.820262712D-15 0.8205225520-15 SADDLE POI NT 35.00000 •0.82 29 047 E-01 +0R- O.OOOE 00 F{X) = EDGEWORTH .' CRAMER SADDLEPOINT 1 SADDLEPOINT 2 0.2071081480 00 0 .1971838330 00 0.333944115D GO 0. 197359615D ' 00 0.21436946OD 00 -0.7295804330-01 0 .2132536550 00 0.2.13783176D CO .0.1523558320 01 0.213614325D 00 0.2137242390 00 ** 0.213702119D 00 0.2I3711194D 00 ... 0.2137072840 00 X = 50.00000 I SADDLEPO INT = 0. 3050665 E-01 +0R- O.OOOE 00 F(X) = EDGEWORTH CRAMER SADDLEPOINT 1 SADDLEPOINT 2 0»658454280D 00 , 0 .656302920D 00- 0.962915067D-01 0.6567731590 00 0.6 8I.7635 54D. OO;" 0*6023925130 .01 0.6814322630 00 0,6 810447900 00 ; -0.1072262050 03 0 .6810769140 00 0.681172 4.06 D. oo "•• 0.681164924D 00 0.68116 1699 0. 00 0.681162475D 00 • = ' .80.00000 SADDLEPOINT. .-F { X } = 0'. 1433714 E CO -+0R- O.OOOE 00 EDGEWORTH CRAMER SADDLEPOINT 1 SADDLEPOINT 2 0.9978666380 00 0.994I95089D 00 0.993002421D 00 0.9 934309 84.0 00 0.9933970090 00 0.987962604D 00 0.992283246D CO 0.993845643D 00 0.993040480D 00 0.9930128950 00 0.9933702620 00 0.99.33789250 00 0.99337.98750 00 0.9933798700 CO Table.IV(c) 53. NON-CENTRAL CHI-SQUARE 1 DEGREE OF FREEDOM NON CENTRALITY PARAMETER =2 N = 40 SADDLEPOINT F('X) 20.00000 •0. 1118 033 E 01 +0R- O.OOOE 00 EDGEWORTH 0.2866515030-06 •0. I37847435D-05 0.251425291D-05 CRAMER 0.500525922D-13 SADDLEPOINT 1 0.2775543660-13 0.274821895D-13 0.274975374D-13 SADDLEPOINT 2 0.266054817D-13 0.275465300D-13 0.2749516700-13 •0. 195930593D-05 0.309407186D-06 0.2749663030-13-0 .2749738830-13 X = 100.00000 SADDLEPOINT •= -0.5825756 E-01 +0R- O.OOOE 00 F(X j '= EDGEWORTH CRAMER SADDLEPOINT .1 SADDLEPOINT 2 0 . 1 58655263D 00 0 .1505020760 00 0.229759540D 00 0. 1505807100 00 0 .1586552630 00 0.322842898D-01 0. 1578734720 00 c .1580584020 00 0.568227400D 00 0. 1579739930 00 0 .157999325D 00 0.1579899980 00 0 .1579918240 00 0.157990658D 00 X = 200.00000 SADDLEPOINT = 0.1298437 E 00 +0R- -O.OOOE 00 F (X) •= EDGEWORTH CRAMER SADDLEPOINT 1 SADOLEPGINT 2 0 .9999683290 00 0.999586453D 00 0.9997311780 00 0.9997457210 00 0 .9998746480 00 0.999757380D 00 0.9997523140 00 0 .9997786290 00 0.9997507550 00 0.999752574D 00 0 .999746572D 00 0.9997525750 00 0 .999751-4710 00 0.9997525750 CO X = 500.00000 SADDLEPOINT = 0.2790024 E 00 +0R- O.OOOE 00 F (X ) EDGEWORTH O.IOOOOOGOOD 01 0.100000000D 01 0.100G000C0D 01 O.IOOCOOOOOD 01 0.1CC0CCO00D 01 CRAMER C.100G0C0C00 01 SADDLEPOINT 1 0.100000000D- 01 0.100000000D .0.1 0.1000000000 01 SADDLEPOINT 2 0.1000000C0D 01 0. 1.000000000 01 O.IOOCOOOOOD 01 0.100000000D 01 0.100000000D 01 3. 6 The Uniform Probability Law. The probability density function and moment genera ting function, respectively, of a random variable distributed uniformly over the.interval (a,b) are f(x) = (b-a)"1 a < x < b 0 otherwise and * r / _i_ \ b t at M(t) = e - e ' (-b-a)t For simplicity, we consider the case a = -b In general, the cumulants, if they exist, may be expressed in terms of the central moments {u^} as a± = Ui , i = 1,2,3 . a5 = U5- " 10MM3 + ^l^ 2 3 a6 = ^6 _ 15wyi2 + lQ4-5(W5+6u2u1+2y1) + 30,^2 For the uniform probability- law, as is easily shown, ui •"=:::• 0 i = 1,3,5, bV(i+l) i = 2,4,6, 55, Thus, •a g = = a6 = 0' b2/3 2 15 b 5 b6 TH b i = -1,3,5, Equation (1.4.6) becomes I', , / cb , -cb\// cb -cbN , — + b (e + e )/(e - e )= x/n To obtain numerical results, for this case, the .last equation v/as solved numerically for c with initial iterate = x/n , and successive iterates obtained by the Newton technique. Note. that since |K^^(t)| < b , a saddlepoint exists only if |x| < nb T • ' 'bt -bt , bt , -bt m. Let. u = e - e and v = e + e . Then the relevant derivatives of the cumulant generating function are K(1  K(3 K<4 K(6 (t) = -t'1 + bv/u. (t) = t"2 - 4b2/u2 (t) = -2t~3 + 8b5v/u3 (t) (t) (t) 6t"4 + 8b4(l-3v2/u2)/u2 -24t-5 - 32b5(2-3v2/u2)v/u3 120t-6 - 32b6(2-15v2/u2+15v4/u4)/u2 iTable Y(a) 5d UNIFORM DISTRIBUTION OVER (A,B) WITH -A=B=2 10 SADDLEPOINT F ( X ) = -IS.00000 •= -0. 1.994526 E 01 +OR- O.OOOE 00-= .0.256647272 E-Q^ : EDGEWORTH' CRAMER SADDLEPOINT I SADDLEPOINT 2 0 . 199 6193 530-04 0 . 1996190530-04 -0 .46629-59280-05 0.4806654010-05 0.2607815870-05 0.2553486560-05 0 .2567846470-05 0.2397507550-05 0.2 5683 00 81D-05 0.2558957110-05 -0.4662.959280-05 0. 1949974560-05 0.2563236030-05 0.256475102D-05 S A DDL F POIM -5.00000 •0 .3899486 E 00 +0R- 0.125F-15 F(.X) .EDGJE^QRTH. 0.867211958 E-01 CRAMER SADDLEPOINT 1 SADDLEPOINT' 2 0.8545 185520-01 Q.854518552D-01 0 .8.396239630-01 0.114245835D 00 0.56 238 62940-01 0.849,3396350-01 0.8726124220-01 0 .36655.17800-01 0.366551780D-01 0 .86716.23800-01 0. 1473192.1 70 0 0 0.8672 57777D-01 0.8673667630-01 0.8672415950-01 2.00000 SADDLEPO.I NT F(X) EDGEWORTH 0. 1 509085',E 00 +0R: 0.705481321 E 00 0.486E-15 .CRAMER SADDLEPOINT 1 SADDLEPOINT 2 0.708058765D 00 0 . 708 1901090 00 0.371477.193D 00 0.707837602D 00 0 . 708058765D 00 0.705 5197 790 00 0 .70 5 5.1 9 7 79D_00. 6.7054803980 00 0/. 245078542D 01 ,-6.1827993540 02 0 .70471.38670 00 0.7054930290 00 ...0 ..7 05.46 3,2.8 9.0 00 0.7054792630 00 X SADDLEPOINT F < X ) EDGEWORTH • 10.00000 0.8983779-<E ' 00 +0R- O.OOOE 00 -0. '..997-5308 27-E-OO —- — -CRAMER SADDLEPOINT 1 SADDLEPOINT 2 0.99691505 ID 00 0 .9 969150 510 00 0 .9.97.49 31720 _00_ 0.907493 172 0 00 0.9975270390 00 0 .997.6713580 00 [0.997318,0520 00 0.9975994960 00 0. 997494130D 00 0.99762 4 8 840 00 0.9975075100 00 ..0.9975302620 00 0.9975296770 00 0.9975305920 00 Table V(b) 2L UNIFORM 01 ST R.I BUT I ON OVER (A,B) WITH -A=8=2 . M = 40 X SADDLEPOINT F ( X ) •= -50.00000 •= -0. 1294131!;E 01 +0R- O.OOOE = 0.107542573 E-12 CO EDGEWORTH CRAMER SADDLEPOINT 1 SADDLEPOINT 2 0 0 -0 .3 78 30 3496D-.3783084960-.61.4 1.9 2.4 500-•11 0 . • 1.1 •11 2426723770-12 0.10890699 2D-;]. 2 0. 107459744D-12 0.107 54 96380-12 0. 10 594 1 8670 0 . 1079364080 0 . 107547^490 -12 -12 -12 -Q 0 .614 1924500-.4455363190-•11 • •11 0. 1 075434?2D 0. 107546058D -12 -12 x SADDLEPOINT •= -10.00000 •= -0. 1892 840.E 00 . +0R- 0.347E-15 F ( X ) = 0.857564838 E-01 ...F.QGFWD.RTH - •• .......C.R A MER .._ SADDLEPOINT I. SADDI. EPO'I NT 2 0 0 .8545185520-.854 51 3552D-•01 0. •01 850771.582D-01 0.114 1274340 00 0.5397668240-01 0.853269240D 0.8589663950--01 -01 0 0 __p. .8575268590-.8575268590->8 5 7 56.5 C.5 3.0-X •01 •01 .01 5.00000 0.149766993D 00 0.8575684]3D 0.857.5724120 -0..857.56.5350D. -01 -01 -01 SADDLEPOINT ' F (X ) = 0.9397053 E-01 +0R- 0.847E-= 0.752531582 E 00 15 EDGEWORTH - . CRAMER "SADDLEPOINTI SADDLEPOINT 2 0 .7532185970 00 o. 7 5.3 2 863680 00 ' • 0 .538637 3090 00 0. 753 16861 CD 00 0 0 0. .7532T3597D .752534940D ...7.5 2 5? 4 9 AO D. 00 00 00__ 0 .•1519990 3 ID 01 -0.475330943D 01 0.752321958D 0 .7525322330 .0..752530474D 00 00 .00 0 . 75253 1572D" 00 0 .7525315410 00 X = 20.GOG00 SADDLE PO INT = 0.3899486;.E 00 +0R- 0.125E-15 FJ.X) = . .0. 997.O6I749..E .00 EDGEWORTH CRAMER SADDLEPOINT 1 SADDLEPOINT 2 0 . 9 969 l-50-'5i?D;--. 00 ••••••• • 0-. 997.1245,1 ID 0 0 . 0.996752300D 00 . 0.9970859110 00 0 . 9 969]50'51'D''00 ,y:- ' 0 .99.7 16 49840 60 - 0.9970539530 00 0 .997059581.0 00. .i., ...l-.P.^q..?.7P0.3.1 8 50 0.0 _. ...0.. 9 97.06 164 70 00 0 .9 9705 9581.0 ]00 '"""•" '.ii • i:L •.'>.'"•'•: }• :\£••'• :'• 0.9 97 061703D 00 o: .99706.16930: 00 *•'•••';''•' • • .0.99706 17450 00 58. Comparison with the exact values for Fn(x) indica tes that the. saddlepoint 2 series again yields the most accurate results (see Tables V (a) and (b)). "\'3.7 Remarks >. In additional test cases (which for the sake of •.brevity are omitted) involving random variables from sections : .3.2 to. 3.6, the"results obtained were qualitatively the same as those reported. Although for the reasons, cited in section 2.2 these methods of approximation cannot be theoretically justified in the lattice case, discrete random variables distributed according to the Poisson probability law were treated. Predictably, the results were erratic and usually inaccurate, but when the argument x was a point of discon tinuity of Fn , the saddlepoint 2 series yielded results ' which were accurate to two significant figures in almost all cases. Only the Edgeworth expansion, when c was.close to . 0 , yielded results of similar quality. 59-CHAPTER 4 OTHER APPLICATIONS. 4.1 The Non-Central Chi-Square Distribution. The form of the characteristic function of the non-central chi-square probability law suggests that an alterna tive approximation to the distribution function of the n-fold convolution of this lav; may be obtained by expanding the integrand in (1.4..4) in powers of \ , where \ is the non-centrality parameter. The objective of this alterna tive approach would be an approximation which v/as useful for very large \ and moderate n Let X-^...,Xn be independent, non-central chi-square distributed random variables, each of whose distribu tion has non-centrality parameter X . The moment genera-ting;.:function of X., (i = 1,2,...) is . -VM(t) = exp[Xt/(l-2t)] (l-2t)"v/2 ! where v = number of degrees of freedom and X = non-centrality parameter. Equation (1.4.4) becomes • 1 - Pn(Xx) = |(l-sign(c)) +~f" [l-2(c+iu)]"Vn/2 60. x exp[-Xx(c+iu) + nX(c+iu)/(l-2(c+iu)"•') ]_du_ (4.1.1). c+iu The saddlepoint c is the root'of the equation Thus, c = $(1 - Vx) . If we write g(z) = nz/(l-2z) , and proceed formally as in Chapter 1, the integral in equation (4.1.1) becomes exp[-Xxc + xg(c)] f [l-2(c+iu)]"nv/2 exp[-x(-g(2)(c)u2/2 — 08 CD + E g<r>(c) (l»)r/rJ)]^H r=3 This can be expanded as a series in powers of X , the asy-mptoticity of which can be demonstrated in a proof very similar to that of Theorem .2.1.2. This expansion, up to the first ! five.terms, is • . .w:.'' -2Xxc2 Fn(>x) = |(1 + sign(c)) -/-!=- th0Q0(p) + X''i(h1Q1(o) + hQb3Q3(p)) + X"1(h2Q2(p) ' + (hxb^ + hQb4) Q4(0) + *h0b|Q6(p)) + X"'5/?(h3Q5(p) . + (h2b3 + hlb4 + h0b5)%(p) + (*hib3 •+ hob3b4)Q7^p) . +'T5hGb3Q9('j)) ^ x"2(h4Q4(p). + (h3b3 + h2b4 + hib5 + hob6)Q6(p) + (*h2b3 + hib3b4 + *hob4 + hQb3b5)Q8(p)  ; + (ihl°l + ihoblHtelo(*\ + ^Thob3Q12(p))^ > (4-1.2) 61. where 0 = cVXg^(c) . Qi(p) (i = 0,1, ...,12) are defined by equations (1.4.21),'(1.4. 22), (1.4.23) and (1.4.24), g(1)(c) x i bi = rm i = * (nx)s ^ (i = 3,...,6) ,. and . 1 i.'Vg^(c)1 h. 0 (i = 0,1,2,... ) , M denoting'the i.'Vg^^c)1 function M ft) = (l-2t)-nv/2 c . For the sake of simplicity, the case y = 1 was considered. Numerical results given below in Table VI indi cate that for moderate values of n and \ , the expansions are nearly equivalent in accuracy and speed of "convergence". As expected, our earlier approximation is.superior where n is large. But even in "extreme cases, such as when n = 1 and X = 1000, the improvement achieved by using the new approximation is very slight. 62. Table VI A • COMPARISON OF APPROXIMATIONS (1. 4. 25 ) AND (4 . 1. 2 )• TO Fp_( \x) X n x EQUATION (1.4.25) EQUATION (4.1.2) 100 15 15 100 1000 8 98 .423374879 .5000000000 .428272116 .427895482 .428277904 .427895482 .428278337 .428280040 .428278338 .428280040 .138175666 .141809322 .. 145448561 .145367769 .145528761 .145535852 .145545801 .145545588 .145546418 .145546215 .369745696 . 373832713 .375295334 .375299472 .375308061 ,375308827 .375308876 .375308896 .375308880 .375308896 4.2 The Doubly Non-Central F-Distribution. 2 2 • Let and x2 'oe ^wo independent non-central chi-square raiidom variables with degrees of freedom f^ and fg and non-centrality parameters X^ and Xg , respectively. The distribution of Xp = ^i^i ^s caHec* the doubly non-~2 X2/f2 central P-distribution. It occurs in the analysis of vari ance and is used in engineering problems where it gives the probability of error in certain communications systems. No simple formula for evaluating the probability integral, F , of Xp is available. Tiku [15] developed several series expansions for F which yield satisfactory approximations when (i) Xj is large and \g is small , (ii) >, 1 is small, and \g is large , (iii) both X-, and Xg are large. •-.. In this section we obtain an alternative to Tiku's approximation for the case when f^ and fg are large and . X-| and Xg are moderate. It is derived by means of the saddlepoint method and is, in part, intended to demonstrate the versatility of this method. Gurland [9] shows that. if X-, and Xg are two independent random variables v/ith characteristic functions and cog > respectively, then the ratio X-,/Xg has a distribution function G which satisfies -l -e T G(x) + G(x-O) '= 1 - —r- lim lim (f + \ V, (t W-tx)d1 n T-» e-»0 -T *V 1 ' d ~~i (4.2.1). -f /2 Putting co,(t) = exp[x,it/(i-2it)](l-2it) J (j = 1,2) L> J we obtain from equation (4.2.1), since F is continuous, 2P(fax/f1) - 1 - K lim lim {J"% / }exprVL- V" T 0eJ^lTT2it TT2xT - fn log(l-2it) - f2 log(l+2xit)]~ (4.2. _- _^ If we. suppose f^ » while X-^Xo and- -g are 'ixed, the appropriate saddlepoint equation is 0 This equation has no finite solution, c . A similar prob lem occurs when we try to expand in terms of -2'^1 °r If the saddlepoint method, is to be useful, there fore, we must suppose that f^ -* <= , fg -+ *> and that X~^ and Xg are fixed parameters. Then c is'the solution of iy-i log(.l-2z) + -| log(l+2xz)] = 0 , which yields c = 2xT.p ~f ^ . Now, - < c < ; thus, formally at least, we can proceed as in section (1.4) and expand the functions in the integrand of (4.2.2) about c in convergent power series. Let f(z) = \ z/(l-2z) - Xpxz/(l+2xz) , • *5. g(z) = - tf± log(l-2z) - | f2 log(l+2xz) * r (2)' u* ^ x(frf2> ,1 1 Then, equation (4.2.2) can be'"rewritten' 1 ffn\ ( \ra en f(J'^(c) ?(xf2/f1) = i(l+sign(c))-- ei{c)+^c)j exp[ s —7—-(iu)J] x exp[ r —rMliu) J exp[ g—JTTTTJ (4.2.3) r=3 Define constants a. (j = 0,1,2,...) by *> • (r) / \ . r ro -j exp[ v SLIM (!{)]. j a r=l a j=0 J a and let bj = g(^(c)/j< , (j = 3,4,...) , and p = c Proceeding as In the derivation of equation (1.4.25), equation (4.2.3) becomes, P(xfp/f1) = *(l+sign(c)) 1-ef(c)+g(c) T. cjf^f^) ~ X ./2ir J=0 . J ~ (4.2.4). Each c .(f ,f,_,) (j = 0,1,...) represents a term of the order (f^^) • The first few terms in the series of equation (4.2.4) are 66. co(frf2) = QQ(p) c1(f1,f2) ='fl Qx(p) +^| Q5(P) i ' h2 c2(f1,f2) =f| Q2(p)+^(b4+a1b5)Q4(p)+^_^ Q6(o) c5(f1Sf2) = f3 Q3(o) + •^(b5+a1b4+a2b5)Q5(p)+i,f-T(b5b4 a a • . a ci|(f1,f2) = *4 Q4(p) + ^(b6+a1b5+a2b4+a3b-5)Q6(p) a a ^R5(|b2+b;5b5+a1b3b4+ia2b|)Q8(p) + -l^^b^. Here, Q^(o-) (i = 0,1,...) is defined as in section 1.4. Numerical results listed in Table VII indicate that good accuracy is achieved when \^ and X2 are small, say X^ < 1 . The expansion is not, of course, uniform in \^ and X2 , and is usally accurate to no more than 2 significant figures for larger . X^ •/ When f= f2 = f , the above approximation can be simplified considerably. The saddlepoint c. becomes -^(1 - -i) . If we letg(z) = X1z/(l-2z)-X2xz/(l+2xz) and h(z) = -\ log(l-2z) - \ log(l+2xz) , then F(x) = -|(l+sign(c)) - 1 eg(c)+fh(c) T. c. (f"*)J (4.2.5). ' Table VII 67. APPROXIMATIONS TG'F-flATIO Nl -NG.PF HTGRFES Of- FREEDOM I'M NUMERATOR N? =N0.OF DEGREES OF FREEDOM T N OF MOM I NA TOR M.t.P.Ml-NQN CFNTRALITY PARAMFTFR IN NUMERATOR N.-C.P. (2 )=MQN CENTRAL! TV PARAMETER TM DENOMINATOR  Nl NZ N.C.P.(l) N.C.P.I?) X F-RATIO 35 40 0.2500 ' 0.3000 0.8000 0.2494549 Qu2511567 . 0.2501 »07 . 0.2 500907 0.2 504148 20 15 0.2500 . 0.3000 0.7500 0 .27641.90 0.2 7796 59 • 0/2753362 0.2725183 0.2745 522 45 '40 0.7000 0.5000 1.0000 0.5000000 • 0,494 96 5 0 0.4949650  0 .495.1 469 . 0.4951469 45 55 1.0000 1.2000 1.0000 0.5000023 0.5011154 0. 501 1153  0.50203.13 •0.502031.3 25 20 I.0000 1.2000 0.8000 0.3059350 0.31 820-83 ' 0.30 7 793 0  0.. 2987 5 78 0.3041053 40 30 0. 1000 0.2000 0.7000' 0.1479470 0..! 48 20 0 7 0 . 1478174 i 0,1474 1.33 0..1477020 25 20 0.2 000 0.1500 0.8000 0.2981952 0.2966938 • ' ' 0 . 2 ° 5 8 7 1 9  ,'•.,:'.. • 0.2954838 "/ v:v'v ' . • ' •• 0.2960377 •- 70 65. • 0.1500 . • •:• 0'. lOO'O 1 .5000 0.9502987 . .'. . .  0.950122.6 ' '"' - ' '."" '• •••'•'• ' '-'-. "• .; '' V ••• • . . • -. 0.9501 31 7  ' : ' • •"  •'• • . 0.950147? 0.9501286 •68-. The-first few coefficients of this expansion are co= ' . c1 = a1Q1(P") + b5Q5(p) c2 = a2Q2(p) '•+ (b^+a-jb^Q^p) + ^^(p) 2 s c5 = a3Q5(p) + (b5+a1b1|+a2b5)Q5(p) + (fca-jb^+b^) : x G^(p) + ^b|<^(p) c4 = a4Q4(p) + (b6+a1b5+a2bJ++a5b5)Q6(p) + (b^+a-jbjb^) + ta2b| + ^)QQ(P) •+ .(fcb|b^.+ ^a-jb^Q-^p) + ^b^O^ p) Here, p = Jfh(2)(c) > (i = 0,1,. .. ) is defined section 1.4, b± = h^fc) (i = 3,4,... ) , and in iWh^(c): a^ (i =0,1,...)' is defined by the equation co (r) / \ , iv v 1 ca exP[ i gy^(rr^-) ] . s a (iy)< r=l r* -Vh^ '(c)7, ' ,,j=0 J. / Numerical calculations listed in Table VIII again indicate - that good accuracy ^is obtained when X^ and Xg are small. For small f , approximation 4,2.5 is considerably more accurate than expansion 4.2.4 for unequal fi of the same order of magnitude. .: . In the case when \^ = Xg = X , a more suitable approximation is achieved for larger. X • say X > 3 ,: if .-.^ :.-^,;.;,-..r. Table VIII .W ; 6a. . APPROXIMATIONS TO F-R AT In ' Nl =NO.OF DEGREES OF FREFDOM IN NUMERATOR N2 =NO.OF DEGREES OF FREFDOM IN DENOMINATOR N.C.P.(1)=NON CENTRALITY PARAMETER.IN NUMERATOR N.C.P.(2)=NON CENTRAL ITY PARAMETER IN DENOMINATOR Nl N2 N „C .P. ( 11 N.C. P. ( 2 ) X F-RATTO  0.3708766 .0....3.6.0.L6.5.2, 0.3627365 0.3634248 0 . 3634509 35 35 5.0000 1.0000 0.7500 0.1490082 , .„.._ Q .1.26J945Z 0 . 1199288 0 . 1236284 ' * . _ 0. 1245618 0.8154488 0... .8.1.5 4 4.8.8. 0.80961.26 0.8096126 0.8096125 10 10 0.0000. 1.0000 2.5000 0.9357405 ...... L_ ... ; _ „ .0 . 93.8AU\3. ... 0.9376194 0.9374207 • 0.9373013 10 10 1.0000 1.0000' 0.7500 . • 0.3253837 „_„„: : " " • • • ' .0 ..33 500.03.. 0.3238 592 . • •' .' V "• • 0.3271418 • '0.3281409 9 0.5000 0.2500 ' 0.6000 0.2176499 , w . *L- —: " • 0.? 163377' •< ' 0.2172729 ' . . \ " 0.2173970 • • ' •• ' - ' 0.2176049 15 15 1 .8000 . 1.2000 ,1.4000 .0.7281970 _*_ _0L. 7.0A8.SL5 7„ 0.7139909 ' ' 0.718 52 32 0.7170775 0.3223013 __.Q...4.2 2 10.6.9 : __. 0.3909000 0.3663807 0.3724224  7 7 0.2500 C.0000 . 0.7875 7 7 • 0.0000 0.0000 2.0000 i 40 40 0.2.500 '" 6.2500 0.7875 70. we expand the integral in powers of• \ * . ,,' 'The appropriate saddlepoint equation now is ' ' d r z • xz n _ ' This equation has root c = (-2L/x + x + l)/( (2,/x)(x-l))': It is seen that - |— < c < -| , • and thus, the functions in the integrand may be expanded about c Again, the approximation is not uniform in the remaining parameters, f^ and fg , hut for moderate values of f , say f^ < 15 > fairly accurate results (3 or more significant figures) seem to be obtained. These and others are tabulated in Table IX. 4.3- Remarks. The results of this chapter suggest that the saddle-point method can be effectively applied to an integral of the ' form [ g(z) e*"h(z^dz , provided that the equation .• -i.e. h^^(z) = 0 has a finite, real solution c . If it does, ..•.v»'- ' f00 , ^hi(u) and the integral can be put in the, form J g^(u) e du , where and . h^ satisfy the conditions of Theorem 2.1.1, :.' this method will-yield an asymptotic expansion in powers :of '• X~'^ Ifas in the example considered in section 4.2, ••'.the problem has other parameters in addition to X , the expansion need not be uniform with respect to them. 71. APPENDIX Al. •' Computing N(z) . . In this appendix is presented a method of evaluating N(z) for complex values of z which was devised by Rubin and Zidek [13]. It uses continued fractional expansions for N and thereby avoids the uncontrollable round-off errors which accrue in using the Taylor's expansion. Their method involves the complex form of Shenton's [14] continued fraction for small values of |z| and Laplace•s continued fraction (see Kendall and Stuart [11], p. 138) otherwise. Since N(z) = 1 - N(-z) , we can without loss of generality assume Re(z)_> 0 . Writing al bl +" a2 ' = • fi_ a2 • a3 . ... b,+ b0+ !>,+• : b2 + a^ / 1 . d > b^ + ... (Al. 1), we obtain, using the Shenton fraction, N(2):;=:l;+n(z) vf-f.^! r£ £ ...) , „,(,).> 0 (A1.2) and, using Laplace's fraction, N(z) =1 -,.n(z)(|- h h h , Re(z) > 0 (A1.3). -2 1 Rewriting equation (Al.2), with t = z , (A1.4). We shall call |_ ^ ,:„ (gn-l)/[(4n-?)(to-l)1 • (n = 1)2)...) (A1.5) the 2nth approximant to the continued fraction in(A1.4). The remainder satisfies ' 2n/[(4n-l)(4n+l)] (2n+LV[ (4n+l) (4n+3) ] * 1- t+ l/8n l/8n l/8n \ ,^\ — ~TT ~TT" • • •, (n •) (A1.6). The continued fraction on the right side of (A1.6) represents the function u(t) which satisfies the equation u = an[1 ~ an(t + u)"1]-1 (A1.7), that is, u(t) = (an - t/2) + [(t/2)2 + 73. (A1.8), where = 1/8 n Let Rn(t) = Re{(t/2)2 + a2} In(t) = Im((t/2)2 + a2} (A1.9) Rn'(t) = t*(Rn(t) + CR^(t) +.I^.(t)]*)]* (n=li2,.. Then (see Ahlfors [2], p.3) ± [R^t) + | In(t)/R;(t)] (R^(t)+0) ' 0 otherwise (A1.10);-. The square root in (Al.lo) has branch points at. + 2ani , and theafunction obtained by choosing either sign is a branch of* the square root. Rather than fix the sign, we take sign[Re(tj/][R;(t) + | In(t)/R^(t) ] , Re(t)*0,R£(t)=}=0 k^t) +| In(t)/R;(t) , Re(t)=0,R^(t)+0 0 ' R^(t) = 0 (Al. 11) t*l(t) + I„.(.t)]* = to obtain a continuous approximation to the continued fraction. 74. Using (A1.8) we obtain, as an asymptotic approxima tion to the continued fraction of (A1.4), z_ 1 2_ (2n-l) 1- 3t+ 5- " , Re(z) > 0 ,(4n-l)(t/2+anW(:t/2)2+a2) (n = 1,2,. . . ) (A1.12). This also gives satisfactory results when Re(z) =0 J. Similarly, we obtain an approximation to the con tinued fraction of (A1.3), fe ^ fe ': Re(z) > ° ' <»-a.5--'> z+ (z +4n) (A1.13). One additional modification is introduced to further improve these approximations. If Hn^z) = J ~(n-l)i exp[-|(t+z)2]dt , Re(z)-> 0 , (n =1,2,...) , (A1.14) and H0(z) - e-2/2 then Hn(z) = (n+1) Hn+2(z) + z Hn+1(z)- , (n = 0,1,...) (A1.15). Letting = Hn_1/Hn yields 75. Qjz) = z + n/Qn+1(z) , (n = 1,2*.'./) (Al. 16). Hence,--Q-^z) > n(.z)/(l-N(z)) = z + 1/Q2(z) 1 2 _(n-ll ^ ^ = 2,3,... ) (A1.17), = z + z+ z+ We- now replace 4n by a^ in (Al.13), where the {a^} , (n = 2,2,...) , are chosen so that the approximation ^(zWz +a^) to Q^2) ' ^s exact at z = 0 , that is, a; = sr2(^+i)/r2(|) , (n-2,5,...) , (AI.IS) where P denotes the Gamma function. 0 Let Rn(z) and In(z) denote Re(z +a^) and 2, / Im(z +a^) . , respectively. Then, if Rk)n(z) = [l(Rn(z)(-l)k-1 + (R^t) + l*(t))*)] (k = l,.2;n = 2,3,. .. ) , (Al.19) we take, as we did in the derivation of (Al.ll), 'sign[ln(z)][ln(z)/2R2^n(z) + i R2jn(z)l Rn(z) < 0, In(z)40,R2^n(z)+0 DRn(z) + iln(z)]* =fn(z)/2R2,n(z> + 1 R2,n(z> >' Rn(z)<0,In.(z) = 0, R2^n(z)+0 Rl,n(Z) + 1 VZ>/2Rl,n(Z> > Rn(z) > 0, Rljn(z) + 0 , S> , R2,n^z^ or Rl,n^z^ = 0 ! n' 76. Summarizing these results we obtain N(z) ~ % + n(z) (jz §--... (2n-l)/[ (4 n-l)(t/2 + l/8n + J(t/2)2 + l/64n2):].) , Re(z) >_ 0 , (A1.21) and • N(z) ~ i-nCzJ^^-fp' • •' 2(n-1)/[z+732+8r2(^i)/r2(-§) ]) , Re(z) > 0 , (Al.22) where the square roots are calculated according to (Al.ll) and (A1.20). The approximations were computed in [13] on a grid for z comprised of 231 points spread over the region D = {z:-5 < Re(z) _< 5,0,_< Im(z) <_ 5} , and Table X was com pleted on the basis of the results. It lists suitable approxi mations for different subregions of D - . For simplicity the approximations in (A1.21) and (A1.22) are denoted by Sr and Cr (r = 1,2,0..) , respectively, where r is a value of n sufficiently large to give an accuracy of at least 10 sig nificant figures over that subregion. TABLE X APPROXIMATIONS TO N(z),, z € D WHICH ARE ACCURATE TO AT LEAST TEN SIGNIFICANT PLACES REGIONS . APPROXIMATION R(Z) . . KB ) : (, 3.5, .. 5 ] V £ °> • 5 ] C10 (-4.25, -3.75 ] -v ; •••/.}. ( 4.25, •; 5 :.' ] - cio (-4.75, -4.25 ] ', .(3.25, 5 ] cio 0.-5, -4.75 ].-•••• .•(. 2.25,. 5 . ] cio ( 2.25, 5 ]' ( o, 5 ] ; °20 (-2.25, -1.75'.'] '..•/••. (3.75, 5 ] c2o (-2,75/ • -2.25 ] : . ( 2.25, 5. ] : c2o (-3-75, ,-2"75 ] ,'v [ . 0, 5 .] ' c2o (-4.25, -3.75 ] [ 0, .. 4.25 •] C20 (-4.75, -4.25 ] : •'[•: 0, 3.25 . ] . C20 [ -5 -4.75 ] • • [. .0, '/•< 2.25 ] C20. (-2.75, -2.25 ] . [ •, 0,' • 2.. 25,' ] C30 ( -75, > 1.75 ] •;•=•• [ 0, ;. .25"-.] ( -25, ..." -75-] • . l • 0, .. 1.75 ] y - s5 (-1.25, -,25 ] . ; [0, .75 ] • : S5 (-1.75, -1.25 ] ; [0, .25 •]. : s5 ( .75, 1.75 ] : ' ( .25, 3.25 ']• : -'• sio (-.25/ -75 ] /$ "•( 1-75, 3.25 ] sio (-1.25,.. -.25 ] ( .75, 2.75 ] sio (-1.75,; ' -1.25 ]'-•'. ^ ( .25/ 2.25 ] sio (-2.25,; -1.75 ] ."• [ 0, 1.25., ] - : sio (1.75, j'•2.25 ] 2.75 . ] ( 1.75, . 2.25 ] •;• < 2-75, .'• 5 ] • si5 ( .75, ; 1.75 ] • •( 3.25, 5 : • ] • Sl5 ( -.25, •'• .75 ] ":; "":'•//. ( 3.25, • 5 ']• : S15 (-1.25,. -.25 ] : , ( 2.75, S15 (-1.75, -1-25 ] " V ( 2.25,' ' ' S15 (-2.25, -i.75 ] (1.25, 3.75 ] S15 '•/:'; ' 78. . A2 Computer Program for Evaluating; the Saddlepoint 2  Approximation. To evaluate the-..saddlepoint 2 approximation (1.4.25), i.a computer program was written in FORTRAN and run on the IBM 360/67 computer using the Waterloo University compiler (WATFOR). The" entire program was written in double precision to keep round off-error to a minimum. Included in this appendix is a listing of a sample run to calculate the approximation (1.4.25) in the case of non-central random variables. Several of the subroutines, such as SUBROUTINE UU, FUNCTION CUMUL, FUNCTION K, FUNCTION KP and FUNCTION KPP, which calculate the constants K. (see J section 1.4), the cumulants, the cumulant generating function and Its first and. second derivatives, respectively, have to be rewritten for different distributions of the variables X^. In addition, the few lines in the main program (lines 47 and 48) which find the saddlepoint are altered with dif ferent cases. A sample set of data cards will contain the fol lowing information: •'; ./• i) Cards 1,2 and 3 - title or comments. ii) Card. 4 - the constant PARA, which may be any parameter that the user1wishes to vary during ... the problem. If PARA _> 1000 , the program terminates. iii) Card 5 - an indicator showing whether the cumu lants are read in (for example, in the case of chi random variables) or whether they are generated by the function CUMUL; the card will read 2 . or 1, respectively. (iv) Card 6 - the constant NCUM, which specifies the number of cumulants to be read in or generated. (v) Cards 6 + 1, 6 + 2,... - (optional) if Card 5 reads 2, the cummulants will have to be read in as data, (vi) Card 7 - the number n ; if n J> 1000 , the program returns to (ii). (vii) Card 8 - the number x ; if x >_ 1000 } the program returns to (vi); otherwise, additional values of x are read in. ... ; ... 80. . . SCOMPILE i DIMENSION ZID(5),Q(13)„G<5),U(6) _2_ COMMON CUM<lO).STDEV.-pARAtNCUM : 3 DOUBLE PRECISION CUM,MU1,STDEV,ENN,PI 1,PI 2,XBAR,U,PARA,X,RT,STAR 4 DOUBLE PRECISION Q,G,W,ABSRHO,SPERR,C,DEXP,OSORT,KAY,K,KP,KPP,FI .. 5 DOUBLE PRECISION.CUMUL,CEE,ESS,ARGUM,ENU,8R,RH0 ,A2,RUTN,ZI0,C0,C1 6 DOUBLE PRECISION TEMP 7 EQUIVALENCE(U(1),XBAR) 8 INTEGER OPT 9 PI1=.3989422804014327 10 PI2=2.D0*PI1 ._1L~_ -..TOL=5 . E-14 „ ;..„.„.::_„ ;.., ' _ .... _ C * c READ IN TITLE '. c 12 READ(5,500) 13 READ(5,501) i _14 . REA0(5,502).. ^ ...... .., . .. .... . 1 15 500 . FORMATC70H 1 ) 16 501 FORMAT(70H 1 ) 17 502 FORMAT!70H ... 1 , V.;,;, \.:_.,;..v„.l,_ ;,'4_,. .lx^A---~~-~ --, -~ ) C c c READ IN VARIABLES •'.••^J'/V\'' 18 22 READ15,400)PARA ; SL*-;vf A: 19 IF{PARA.GT.999.:)G0T0. 20 :; . ;' . 20 READ(5,100)0 P ^:lz^::;:^iili:v-.:..- .; ^....l:;..:;!^:;: t.. u - ....... : 21 100 f '. FORMAT!I 3) K. C IF OPT={1,2).CUMULANTS ARE*(GENERATED,READ IN) 22 C r READ(5,100)NCUM , - U .... C r MAXIMUM NCUM = 7 23 IFIOPT.EO.nGO TO 1 24 REA0(5,300) (CUMU ),1=1,NCUM) 25 300 FORMAT 13026.16) ; ,26 GO TO 222 _ '• . —. ......... ... 27 1 DO 3 1=1,NCUM 28 3 CUM fI) = CUMUL(I) / 29 77? RFADC5.1071N 81. 30 WRITE t6,499) , . . 31 499 F0RMAT( 1H1) ' r' •; 32 WRITE(6,500) 33 WRITE (6,501) " ' ' '0~. 34 WRITE16,502) 35 WRITE(6,900)N 36 900 FORMAT<//34X,3HN =,I3//> 37 • C 102 FORMAT(15) .. .. V. c • r MAXIMUM N = 9999 38 IF(N.GT.9999)G0 TO 22 39 c 21 READf5,400)X ; _C. c ........ . MAXIMUM X = .999. .„ ...... . 40 v. IF(X.GT.999.)G0 TO 222 41 400 FORMAT<F10„5) C c c START OF INITIAL CALCULATIONS 42 MUl=CUM(i) 43 STDEV=DSQRT.CUM(2)) 44 ENN=N 45 W=(X-ENN*MUl)/(DSQRTlENN)*STDEV) 46 XBAR=X/ENN ' C f FIND SADDLEPOINT 47 RT=o2 5D0/XBAR**2+PARA/XBAR 48 C=.5D0*(1.-.5D0/XBAR-DSQRTIRT)) 49 SPERR=0. . 50 _ 73 DO 12 1 = 2,6. ... ' ... 51 12 U(I) = 0. 52 CALL UUIU,C) 53 STAR=DSQRT(U(2)) 54 ARGUM=-C*X+ENN*K1C) 55 KAY=DEXP(ARGUM) *PIl , 56 . r ENU=ENN*U(2) . . .. \* c c PROCEEO WITH SECOND SADDLEPOINT APPROXIMATION 57 23 RHO=C*DSQRT(ENU) . / 58 A2=RH0**2*.5D0 -59 . _,RO=RHO .__ _._ .„ :.. . 60 ABSRHO=ABS(R 0) 61 IFtABSRHO-2.25)38,38,37 62 37 Q(1)=CEE(30,ABSRH0)*SIGNUM(C> 63 GO TO 39 64 38 Q(l)=(DEXP(A2)/PI2-ESS(20,A8SRHO))*SIGNUM(C) 65 39 FACT=I. .. 66 DO 10 1=1,11,2 67 FACT=-1.*FL0AT(I-2)*FACT 68 Q(1+1)=FACT-RHO*QlI) 69 10 Q(I+2)=-RH0*Q(I+l) 70 G(1) = QU) 71 G(2)=U(3)*Q<4)/(STAR**3*6.D0) . . 02 G(3)=U(4)*Q(5)/tU(2)**2*24.D0)+U(3)**2*Q<7)/<U( 2)**3*72.D0) 73 G(4)=Ut5)*Q<6)/(STAR**5*120.00)+UC3)*U(4)*Q(8)/(STAR**7*144.D0)+ 1U(3)**3*Q(10)/(STAR**9*1296.D0) •V. •82. 74 G(5)=U{6)*Q(7)/IUi2)**3*720.D0)+lU(4)**2/1152.D0+U(3)*U(5)/720.D0) i*CM9)/U.2)**4*U(3)**2*U(4)*Q(11)/{U(2)**5*1728.DO)+U(3)**4*Q(13)/( 2U{2)**6*31104.D0) 75 RUTN=l./DSQRT(ENN) . . 76 ZIDU> = .5D0*<1.+SIGNUM(C>)-KAY*G<1> TL PJL_U 1 = 2 v 5 : . '. 78 11 ZID(IJ=ZID(I-1)-KAY*G(I)*RUTN**{I-1) C . C . PREPARE OUTPUT . 1 _ .:• _ .. „ ... C 79 XD=X 80 CS=C 81 SPE=SPERR 82 WRITE{6,1000)XD,CS,SPE 83 1000 FORMAT{10X,1HX,11X,IH=,F10.5/10X,13HSADDLEP0INT. =,E16.8,2X,4H+0R-, 1E10.3/10X,4HF(.X),8X,1H= /) 84 WRITE(6,1100)(ZlOCIltl'lf5) 85 1100 FORMAT(4X,13HSADDLEPOINT 2/5(1X,D16.9/) ) 86 GO TO 21 087 20 CONTINUE . 88 STOP , • : 89 END •/ . ' ( 90 FUNCTION SIGNUMIT) ... • 91 r....1 ' .. ' I-.DOUBLE PRECISION T . ' 92. i..:ji,'.l.,..'.f.,.-..,.'J J,..-. IFIT) 1,2,3 ;.•.,.„„ : ........... ' 93 ' ' 1 SIGNUM=-1. 94 GO TO 4 95 2 SIGNUM=0. 96 GO TO 4 97 3 SIGNUM=1. 98 .4 .RETURN....:. ' '. \::: .. ....... . . .... _ 99 END 100 f DOUBLE PRECISION FUNCTION FKT) C .. NORMAL. DISTRIBUTION FUNCTION . ... 101 C DOUBLE PRECISION DEXP,DSQRT,ESS»CEE,A,FACTOR.FF,P11,T,TT 102 PI1=.7978845608028654 103 A=-.5D0*T**2 / 104 FACTOR=.5D0*DEXPtA)*PI1 105. . , ......... ,.TT=T . ... ... . , . 106 TSP=TT 107 A=T 108 IF(TT)6.7,8 109 7 FF=.5D0 110 GO TO 15 111 •_. 6 A=-T :,. .... .' 112 8 IF{ABS{TSP)-1.75)1,1,2 113 1 FF=.5D0+FACT0R*ESS18,A) U4 GO TO 5 115 2 IF<ABSlTSP)-2.2 5)3,3,4 116 3 FF=.5D0+FACT0R*ESS(13,A) 117 GO TO 5 . . . 118 4 FF=1«J-FACT0R*CEE(25,A) 119 5 IF(TT)9,15,15 120 9 FI=1„-FF 121 GO TO 16 122 15 FI = FF 123 16 RETURN .., 124 END 125 DOUBLE PRECISION, FUNCTION ESStN,Z) C C SHENTON CONTINUED.FRACTION C 126 DOUBLE PRECISION Z,DENOM,BR,RUT,T,EN,DEXP,DSQRT 127 ESS=0. . .  128 IF(Z.EQ.O.)RETURN 129 EN=N : 130 T=l./Z**2 131 RUT=.25D0*T**2+1./(64.D0*EN**2) 132 : MULT=4*N-1 133 NUM=2*N-1 . . :  134 SIGN=-l. •• 135 DENOM=FLOAT(MULT)*{.5D0*T+.125/EN+DSQRT{RUT)) 136 LI M=NUM J.'..'. • ' ... . 137 DO 1 1=1,LIM 138 MULT=MULT-2 139 DENOM=FLOAT(MULT)*< (SIGN»1. ) »T-S IGN»1. ) *. 5D0-t-SI GN*FLOATt NUM) /DENOM 140 NUM=NUM-1 141 1 SIGN=-SIGN 142 ESS=Z/DENOM . _ . 143 RETURN 144 END 145 DOUBLE PRECISION FUNCTION CEEJN,Z> C C LAPLACE CONTINUED FRACTION 146 DOUBLE PRECISION DEXP,OSQRT,Z,DENOM,RUT,AI,A2,GAMM 147 Al={ FLOAT{N3 + 1. ?/2.D0  148 A2=FLOAT<N)/2„D0 149 RUT=Z**2+8.*{GAMM(A1)/GAMM(A2))**2 150 DENOM=Z+DSQRT(RUT) . . 151 LIM=2*N-2 152 DO 1 1=1,LIM : 153 NUM=2*N-1-I ; : ; :  154 1 DENOM=Z+FLOAT(NUM)/DENOM 155 CEE=1./DENOM 156 .„ RETURN ... - * . 157 END 158 DOUBLE PRECISION FUNCTION GAMM(X) C C GAMMA FUNCTION C 159 DOUBLE PRECISION X,XX,FACT 160 IF ( X.LE. 1. )G0 TO 10  161 N=X 162 XX=N 163 IF(XX.NE.X)GO TO 2 164 FACT=1. 165 N1=N-1 L66 DO t 1 = 1.Nl  84 167 1 . . FACT=FACT*FLOAT (I ) . 168 GAMM=FACT 169 GO TO 11 170 2 LIM=2*N-1 171 FACT=1. 17? DO 3 J=1,LIM.2 F 173 3 FACT=FACT*FLOAT(J) 174 GAMM=FACT*l.77245385O?O5516D0/2.D0**N 175 — „. GO TO 11 - ._ _ .. .,.„„ • 176 10 WRITE16,100) 177 100 F0RMAT(5X,20H ERROR IN GAMMA FCN. ) 1 78 11 RETURN 179 END 180 r SUBROUTINE UU( - U,C) . c DERIVATIVES OF C.G.F. CALCULATED AT C 181 c DIMENSION U(6) 182 . COMMON CUM110),STDEV,PARA,NCUM 183 DOUBLE PRECISION CUM,STDEV,U,RT,PARA,CAL,DSQRT,C 184 RT=.25D0/U(1)**2+PARA/U(I) 185 CAL=o5D0/U(l)+DSQRT(RT) 186 U(2)=2./CAL**2*(i.*2.*PARA/CAL) 187 U(3)=8.D0/CAL**3*ll.*3.D0*PARA/CAL) 188 U(4)=48.D0/CAL**4*(1.+4.D0*PARA/CAL) 189 U(55=384.D0/CAL**5*(l.+5.D0*PARA/CAL) 190 U(6) = 3840.D0/CAL**6'M l.+6.*PARA/CAL) 191 RETURN 192 END -193 c DOUBLE PRECISION FUNCTION CUMUL(J) " C CALCULATION OF CUMULANTS 194 C COMMON CUM(10),STDEV,PARA,NCUM 195 - DOUBLE PRECISION CUM,STDEV,PARA,UMUL_ J _ 196 UMUL=1.+ PARA: 197 JMl=J-l 198 FACT=1. 199 I F ( J-1) 1, 1, 2 / 200 2 DO 3 K=1,JM1 201 _ ......3. FACT=FACT*2.*FLOAT{ K) . . _;*. _  ; 202 UMUL=FACT*{1.+ PARA*FLOAT(J)) 203 I CUMUL=UMUL 204 RETURN 205 END 206 c DOUBLE PRECISION FUNCTION K IS-) C CUMULANT GENERATING FUNCTION (C.G.F.) j 207 C . COMMON CUM(10)»STDEV,PARA,NCUM 208 .... DOUBLE PRECISION CUM,STDEV,S,PARA,ARG.DLOG . ' 209 IF(S-.5)1,2,2 210 2 WRITE(6,100) ?ll 100 F0RMAT(5X.13H K UNDEFINED ) 212 K=0. 213 GO TO 5 214 1 ARG=1„-2«.*S 215 K=-.5D0*DL0GlARG)*PARA*S/ARG 216 5 RETURN 217 END 218 r DOUBLE PRECISION FUNCTION KP (S) . c c FIRST DERIVATIVE OF C.G.F. • ' ! ' 219 COMMON CUM(10),STDEV,.PARA,NCUM 220 DOUBLE PRECISION CUM,STDEV,S,PARA, ARG 221 .... IFIS-.5)1,2,2 222 2 WRITE(6,100) 223 100 F0RMAT(5X,13HK UNDEFINED ) 224 K=0. 225 GO TO 5 ( 226 ARG=1.-2.*S 227 KP=lo/ARG+PARA/ARG**2; . 228 RETURN .'. ." 229 " I' . END ';: VV 230 r DOUBLE PRECISION FUNCTION KPP(S) c r SECOND DERIVATIVE OF C.G.F. 231 .v. COMMON CUM(10),STDEV,PARA,NCUM 232 DOUBLE PRECISION CUM,STDEV,S,PARA, ARG 233 IFIS-.5)1,2,2 234 . 2 WRITEJ6,100) .. . • 235 100 F0RMAT(5X,13H K.UNDEFINED ) 236 K=0. 237 GO TO 5 238 1 ARG=1.-2.*S 239 KPP=2./ARG**2+PARA*4./ARG**3 240 5 RETURN . ; _ _ 241 END / 86. ' REFERENCES [1] Abramowitz, M. and Stegun, I.A., Handbook of Mathematical  Functions, Dover Publications, Inc., New York (1965). [2] Ahlfors, L.V., Complex Analysis, McGraw-Hill Book Company, Inc., New York (1953). [3] Cramer, H., Mathematical Methods of Statistics, Princeton University Press (1945). [4] Cramer, H.', Sur un nouveau theoreme-limite de la theorie des probabilites: Actualites Scientifiques et Industriel-les, No. 736, Hermann et Cie, Paris (1938). [5] Daniels, H.E., Saddlepoint Approximations in Statistics, Ann. Math. Stat., vol. 25, pp.631-650 (1954). [6] De Bruijn, N. G., Asymptotic Methods in Analysis, North-Holland Publishing Co., Amsterdam (1961). [7] Feller, W., An Introduction to Probability Theory and its  Applications, Vol. II, John Wiley and.Sons Inc., New York (1966). 7 [8] Gnedenko, B.V. and Kolmogorov, A.N., Limit Distributions  for Sums of Independent Random Variables, Addison-Wesley Publishing Co. , Cambridge, Mass. (1954). [9] Gurland, J., Inversion formulae for the distribution of ratios,- Ann.,.-Math. Stat., vol. 19, pp. 228-237 (1948). [10] ' Jeffreys, H. and Jeffreys, B.S., Methods in Mathematical  Physics, Cambridge University Press (1950). 87. [11] Kendall, M. G. and Stuart, A., The Advanced Theory of  Statistics, Vol. I, C. Griffin and Co., London (1958). ['12] Parzen, E., Modern Probability Theory and its Applica tions, John Wiley and Sons, Inc., New York (i960). •[13] Rubin, H. and Zidek, J. , Approximations to the Distri bution Function of Sums of Independent Chi Randoirr Vari ables, Technical Report No. 106, Dept. of Statistics, Stanford University, Stanford, California (1965). [14] Shenton, L.R., Inequalities for the normal integral including a new continued fraction, Biometrika, Vol. 4l, pp. 177-189 (1954). [15] Tiku, M.L. Series expansions for the doubly non-central F-distribution, Australian Journal of Statistics, Vol. 7, pp. 78-89, Sydney (1965). [16] Watson, G. N., Theory of Bessel Function, Cambridge Uni versity Press (1948). 

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