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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 i n the Department of MATHEMATICS We accept t h i s thesis as conforming to the required standard. The University of B r i t i s h Columbia In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o lumbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u rposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f M* rH£Mft-Tt-C$ The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date SePT.'&i*, i i . 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 i t s parameters. i i i . 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 20 CHAPTER II. THE SADDLEPOINT APPROXIMATIONS 21 ' • 2 . 1 • Asymptotic Expansions ' 21 2 . 2 ..The. L a t t i c e Case • 33 CHAPTER II I . COMPUTATIONS 37 3 . 1 ' Remarks on the Tables " 37 3-2 " Chi Random Vari a b l e s ; " 39 3 . 3 The Exponential P r o b a b i l i t y Law 43 3 . 4 The Normal P r o b a b i l i t y Law 49 3 . 5 ' The Non-Ceritral Chi-Square Pr o b a b i l i t y . 49 Law . . . . 3 . 6 The Uniform P r o b a b i l i t y Law 54 3 . 7 Remarks . 5 8 CHAPTER IV. OTHER APPLICATIONS 59 • 4.1 . 'The Non-Central Chi-Square D i s t r i b u t i o n 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- t i n g the topic of t h i s thesis and for generous assistance given during i t s . w r i t i n g . The f i n a n c i a l support of the National Research Council of Canada and of the University of B r i t i s h Columbia i s also g r a t e f u l l y acknowledged. INTRODUCTION . It i s often necessary to approximate the d i s t r i b u t i o n of a s t a t i s t i c whose exact d i s t r i b u t i o n i s unknown or cannot conveniently be calculated. For example, i n the evaluation of error p r o b a b i l i t i e s i n some types.of communications systems or sometimes i n the determination of the power function of a 1 n l i k e l i h o o d r a t i o test, the p r o b a b i l i t y 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 t h i s type of problem i s due to Edgeworth ([3L pp. 2 2 8 - 2 2 9 - [ 7 ] , p. 5 1 5 ) . Another i s that of Cramer ( [ 7 ] , 'p. 5 2 0 ; [ 4 ] ) , which was designed to provide an accurate approximation even i n the case where x i s permitted to depend on n When the c h a r a c t e r i s t i c function of the s t a t i s - t i c i s known, in v e r t i n g the Fourier transform e x p l i c i t l y i s often impossible, while a numerical integration routine i s z too time-consuming for procedures requiring high accuracy, e s p e c i a l l y when U ( t ) | i s not r a p i d l y decreasing as |t| - » . In t h i s thesis we consider two approximations, sug- gested by Rubin and Zidek [13], to the d i s t r i b u t i o n function of : the sum of n independent random variables'. Each of these approximations i s c a l l e d , i n [ 1 3 ] , a saddlepoiht approximation in keeping with the terminology used by Daniels [5], who seems to have been the f i r s t to introduce into the l i t e r a t u r e of p r o b a b i l i t y approximation theory the method upon which the approximations i n [ 13] are based. Whereas i n t h i s thesis we are concerned with problems r e l a t i n g to d i s t r i b u t i o n functions, the work of Daniels pertains to density functions. The e f f o r t s of Rubin and Zidek [13] are directed toward the problem of f i n d i n g an approximation to the d i s t r i - bution function of ( | Z-ĵ  | + . . . + | Z | ) , where the {Zi} are independent random variables and each Z. ( i = l,...,n) i s normally di s t r i b u t e d with mean 0 and variance 1 Each of th e i r approximations i s roughly equivalent, i n terms of required computer time, to those of Edgeworth and Cramer. In [13] i t i s shown for the problem considered there, on the basis of numerical r e s u l t s , that one of.the approximations i s superior to either of the two c l a s s i c a l methods. Even for the case n = 10 , where for values of the arguments con- sidered, the older methods y i e l d an accuracy of at most two s i g n i f i c a n t f i g u r e s , i t gives r e s u l t s accurate to f i v e s i g n i - f i c a n t figures. In Chapter I of t h i s thesis a l l of the approximations mentioned above are presented. .Also given i s an inversion f o r - 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 i n [13] are i n fa c t asymptotic ex- , pansions. In [13] i t i s suggested that these r e s u l t s might be obtained by using an adaptation of the argument given by Daniels [5], which i s based on the method of steepest descent. Here we.give f o r the case of non-lattice random-variables 3. simple d i r e c t proofs which use Laplace's method for integrals and special features of the present problem. The re s u l t s are, in one case, an expansion i n powers of n *- , and, in the other, as i n Daniels' case for densities, an expansion i n powers of n""*" The approximations are then compared, i n Chapter J>, with those of Edgeworth and Cramer for a vari e t y of spe c i a l cases on the basis of numerical computations for small values of n . The res u l t s are q u a l i t a t i v e l y the same as those found i n [13] for the spe c i a l case considered there, which i s presented for completeness i n section ( 3 . 4 ) . The r e s u l t s given i n Chapter 3 are obtained with the intention of comparing the various approximations described j- • e a r l i e r . Except for the cases considered i n sections 3.4 and 3.7, the desired values can be found with reasonable accuracy from e x i s t i n g tables. In Chapter 4 , some additional applica- tions-of the "saddlepoint method"'are considered. • These i n - volve the non-central chi--square. d i s t r i b u t i o n f or large values of i t s non-centrality parameter,; :and the doubly non-central F - d i s t r i b u t i o n for selected s p e c i a l cases defined i n terms of i t s four parameters. In the l a t t e r case p a r t i c u l a r l y , e x i s t i n g tables are inadequate, and the saddlepoint approxi- mation may be of p r a c t i c a l value. Two appendices are supplied. In the f i r s t i s given a method of evaluating the normal d i s t r i b u t i o n function for either r e a l or complex values of i t s argument. The method, which uses continued f r a c t i o n s , i s given i n [ 13 ] . In. Appendix 2, we describe a program written i n 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, i d e n t i c a l l y d i s t r i b u t e d random v a r i a b l e s , each having d i s t r i b u t i o n funct ion 2 F , density funct ion f , mean u = 0 , variance a , and c h a r a c t e r i s t i c funct ion cp ; that i s , co(t) = E [ e l t X ] (-»<t<«0 = J " e i t x dF(x) , where i = A/^T ¥ e assume the moment generating funct ion of X^ exis ts on a non-degenerate i n t e r v a l (a,b) and denote i t by M ... Let K be the cumulant generating funct ion . Then M(i t ) = cp(t) , / - ~ < t < » , and • • K(t) = l og 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 d i s t r i b u t i o n and density functions of the stan- dard normal d i s t r i b u t i o n w i l l be denoted by N and n , re s - pectively. . ' Let P denote the p r o b a b i l i t y d i s t r i b u t i o n function n n of E X . / ( J ? T G ) , (n = 1 , 2 , . . . ) ; that i s , i = l 1 where . P denotes the , n r-f old .convolution • of P . . More generally, i f : u ̂  0 , F n and P. w i l l denote the d i s t r i b u - n t i o n functions of E (X. - u ) / (,/n a) and (X. - ̂ ) , i= l 1 1 respectively, while M w i l l denote the moment generating func- t i o n of (X i - 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 i n x , for some polynomials R-̂ ,. . . ,Rr , _e ach depending on u-̂ ,. . .u , the f i r s t r moments of F , but not on n , _or otherwise on P _or r PROOF. See F e l l e r [7 ] , p. 515. The series i n (1.2.1) i s known as the Edgeworth expansion of F . The construction of the polynomials, n R k (k = l , . . . , r ) , i s described i n F e l l e r [7] , p. 509. With 7. I t s . f i r s t few. terms given explicitly, this expansion for F n is F n(w n = N(w n) - r T * r i f N( 5 )(w n)] + n - 1 [ ^ N ^ ) ( w n ) + ^ x | N < 6 > ( w n ) ] 3 r 5 I iV ^ n ; T 77 '"3*4 ^ wn ; ~"9J"'\5 ? x f i ^ 2 N ( 8 ) ( W ) + 2100 ,2, „(10), v , 154 00 \4 ..(12 N "TOT X3 X4 N l ;( wn) + -121- *3 N } (w j ] - (1.2.2), where wn = (x - nu)/(Vn a) , \ n = ctn/an ( a n denoting the n t h cumulant of F ), and denotes the i t h derivative of N . In practice, i t 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 i t then tends to give negative values for F n ( 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) w i l l f a i l to hold. Since i n t h i s case both • F n ( x ) and N(x) converge to 1 as n -» » , a more appro- p r i a t e c r i t e r i o n of the accuracy of the approximation i s the r e l a t i v e error. In p a r t i c u l a r , we would li k e . t h e r e l a t i o n 1 - F n ( x ) (1.3.1) 1 - N ( x ) . " to hold when both x and n tend to i n f i n i t y . This r e l a t i o n i s not true generally, since, for example, i n the case of the symmetric binomial d i s t r i b u t i o n , 1 - F n ( x ) = 0 f o r a l l x > Jn . The l i m i t i n (1.3.1) does hold i f x = o(n 1 / /^) , t h i s being a consequence of the following more general r e s u l t . It i s due to H. Cramer [4] and was generalized to variable components by F e l l e r ([7], p. 524). Let the function V be defined by the equation • \ z 3 X(z) = K(h) - h K ^ ^ h ) + -|z 2 (1.3.2), where- h i s obtained as a power series i n z by inv e r t i n g the series oz = V : ( r r i ) . hr"1 (1-3.3) r=2 = (h) Equation (1.3.2) implies X(z) =• X 5/6 + ( X 4 / 2 4 - \|/8)z + (x^/120 - \j\h/12 + \j/8)z2 + . . . • • . . ( 1 . 3 . 4 ) THEOREM. 1 .3 .1 (Cramer). Suppose there exists a 00 _ number h Q such that j e dF(x) exists for a l l — CO h e (-h0,h ) . Let x _be a r e a l number, which may. depend on n , such that x > 1 and x = o(n2~) as n - « Then v , 1 - F n ( x ) « [1 - N(x)] exp [£- HJL-mi + 0 ( ^ ) ] m^y'* (n - .) ( 1 . 3 .5a) For x < -1 , the corresponding r e l a t i o n i s F n ( x ) » N ( x ) exp [j^X ( ^ ) ] [ l + 0 ( ^ ) ] , (n - ' ( 1 . 3 . 5 b ) PROOF. See F e l l e r [ 7 ] , p. 517. When the Cramer approximation i s applied, d i f f i c u l - t i e s may be encountered i n the inversion of the series ( 1 . 3 . 3 ) * because the series given i n ( 1 . 3 . 4 ) may f a i l to converge. Examples where.this occurs are given i n section 3-3. The ap- proximation can s t i l l be applied i n these cases i f i t i s pos- s i b l e to invert ( 1 .3 -3 ) a l g e b r a i c a l l y . 1 0 . 1 . 4 The Saddlepoint Method. We now present two saddlepoint approximations to F n obtained by Rubin and Zidek [ 1 3 ] . They are based on an inversion formula derived i n Lemma 1 . 4 . 1 from the Gurland [ 9 ] inversion formula. It asserts that -e T .. - i x t 1 - F ( ) = * + lim lim { f + f } M n ( i t ) d t (1.4.1) LEMMA 1 . 4 . 1 . Assume that the moment generating function M exists i n the region (z : a < Re(z) < b] . Then . p-e .T Y - i x t s e i o:u T + u ^ M ( i t ) d t = W - I_l e-< c + i u> x M V + l u ) ! ^ - t sign ( = ) • ' (1.4.2)" for every r e a l 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 l i n e i n t e g r a l i n the. complex plane of the. function 11. f ( z ) = e " Z X M n(z)/(27riz) along the contour I = I 1 I 2 +...+ Ig where, for fixed p o s i t i v e 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 - 2 J Then f f(z)dz = 0 . J I Now, . c • i t - f ( z ) dz | f | exp (-x(y+iT)) M n(y+iT) | dy J I 2 ^ J o • |y+iT since ' • ' / ' • •/ [^(y+iT) |; = |E(exp[(y+iT)(X 1 + . . . + X n ) ] ) | < E(exp[y(X 1 + ... + X n)]) = ^ ( y ) • Hence, If f( 2)dz| <.1 t ^ - | ( l - e" c x ) ] , 2 where A = max M n ( y ) , and therefore o<y<c 12. lim f f ( z ) dz = 0 Sim i l a r l y , lim j f(z)dz = 0 T— i 4 Consider J f(z)dz . By the residue theorem, x6 i , v f(z)dz = 1 :(e) v where c(e) i s a c i r c l e with centre at the o r i g i n , radius' e , and the i n t e g r a l i s taken i n a counterclockwise direc- tion. ~ We s h a l l now show that TT/2 . fl . . -77/2 . a lim f f ( e e i e ) e i e 1 B dfi = lim f ' f ( e e i e ) e i e i p dfl :e-o\ J -TT/2 e-o TT/2 (1.^.3) Prom t h i s i t follows that lim f(z)dz = \ , and the desired r e s u l t i s an immediate consequence. Now, TT/2 f e - X € ( c o s 9 + i s i n e> M n(e cos fi+e i s i n e ) d -TT/2 1 f ' -xefcosP + i s i n 9 ) w n / „ , . . - 7 5 = e v ;M (e cose + e 1 sine ) d e | W J TT/2 1 TT/2 = |W f [ - e - x e ( c o s f i + 1 s i n * ) M n ( e cose +-e i sine) J - T T/2 13. _ e-x€(-cose + i s i n e ) M n ( _ £ c o s 0 + g ± s i n s ) ] d 0 | < 1_ J V 2 | e - x e c o s 9 M n(e cose + e i s i n 9 ) - e X e C O s e -TT/2 x Mn(-e cose + e i.sinp)|de By;the Legesgue bounded convergence theorem, the l a s t quan- t i t y converges to 0 as . e -• 0 and the r e s u l t , (1.4.3), i s established. Combining the re s u l t s 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 i s similar. We s h a l l make use of the inversion formula (1.4.2) i n the form, 1 l' l C = 1 - F N ( ^ x/a) - -1(1 - sign(c)) + TJ^- .J exp[-nx(c+iu)+n logM(c+iu)] — 00 where we s h a l l choose c to be a saddlepoint of the exponent i n the i n t e g r a l ; that i s , c i s 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 i n t e g r a l i n (1.4.4) w i l l be understood to mean that of the Gurland inversion formula. Daniels [5] showed that (1.4.6) has a single r e a l root under f a i r l y general conditions. 14. THEOREM 1.4.1 (Daniels). Assume that (t) = e K ^ = f e t x dP(x) M converges for r e a l 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 b o are f i n i t e i f and only i f K(t) exists for a l l r e a l t , and (1.4.6) has no r e a l root whenever x i [a,b] , ( i i ) f or 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 , ( i i i ) f or every x e (a,b), where ?a and b may be i n f i n i t e , there exists a corresponding c i n (-c-^Cg). i f lim K ^ ^ t ) = b and (1) t-»+c2 lim Kv (t) = a (these conditions are s a t i s - ' t — c ^ f i e d automatically unless a or b _is i n - f i n i t e ) . ' PROOF. See Daniels '[5]. Since K(t) converges for -c-^ < Re(t) < c 2 , 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 a l l 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) ( i u ) r (1.4.7) r=3 rJ Let a* - [K<2> ( C ) ] * , ' b r = K<r> (c) i r / ( r J a * r ) a r = ( - i ) r / ( c a * ) r , (1.4.8) K(c,n,x) = J2T e " n x c + n K< c> , I(x,n) = ^ J " E - N X ( C + I U ) + N K ( C + I U ) du c+iu Then, proceeding formally, cm - n r lh K ; ( c ) ( i u ) -na u I(x,n) = K( c,n,x) J e" v r.' e 2 ~ ./2TT du -» c+iu ca ^ K l c ^ x ) j ( l + C f f ; ^ ) e r=3 i y ,-1 n v b r(y/,/rI) r _y2/2 dy ca ./27m (1.4.9) 16. Now, / _ _ i y _ _ v - i » 2 \1 + C a * / j j ) = 1 +' I a r• (y//n) IyI < c a V n , • • ( 1 . 4 . 1 0 ) and eo vr n i b (y/./n) _^ • 6 e r=3 = 1 + n 2 b^ y^ + n 1 (b^y 4 + # b^y ) + n " V 2 ( b 5 y 5 + b 5b 4y 7 + £ b^y 9) + n-2(.b6y6+[i-bf+b3b5]y8-^b2b4y104ifb4y12 + . . . ( 1 . 4 . 1 1 ) . Hence, _^ r=3 */n m=0 ,/n ( 1 . 4 . 1 2 ) , where ; d 0 ( y ) = 1 • ' ! ' 2 4 2 6 d 2(y) = a 2y + (b 4 + a-jb^y + | b^ y d^(y) = a 4y 4 + (b 6 - K a . ^ + agb^ + a^b^y 6 + (|b2 + b ^ + a±b3bk + t a 2 b 2 ) y 8 + ( ^ b 2 + ̂ b ^ y 1 0 + b^y 1 2 M :SM>.\; . v : } K . . •;•'• , ; '•''•; ( 1 . 4 . 1 3 ) , and, in general, d 2 k - l ^ y ^ i s 3 X 1 o d d polynomial in y , while:•d 2 k(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 w i l l be shown in the following chapter that • 17. 2m I(x,n) ~ K ( j;*"'*) T, d V̂ =; (1.4.14) i s an asymptotic expansion. Here, d2m = f n ( y ) d2m^) ^ (1.4,15) — CO 1 _ 2 / 2 ( r e c a l l that n(y) = e ^ ) , and the series i n (1.4.14) i s obtained formally by interchanging summation and integration i n the expression obtained from (1.4.9) by replacing the f i r s t two factors i n i t s integrand by t h e i r series expansion. The f i r s t few c o e f f i c i e n t s i n (1.4.14,) are d 2 = a 2 + 3(b 4 + a-jb^) + b | (1.4 .16) d4 = 3a^ + 15(bg + a-ĵ b̂  + a 2b^ + a^b^) + 105(tb i | + b^b^ + a 1b 3b 4'+ |a 2b|) +'945(t'b4b| + ^ b ^ ) + b^ . E x p l i c i t l y , equations (1.4 .16) are, with K^r^ (c) = K° (r ;=• 1,2,...) , do = 1 : d 2 = - ( c a * ) " 2 + a * - 4 ' ^ - K 5) - -IT 4 ( o * r 6 (1.4.17) , d4 = 5(ea*)- 4 + | a * " 6 ( - ^ Kg + ^ K 5 - ^ + ^ c c 35 * - 8 , l K2 1 1 „ 1 2v ,35 *-10 + "2T a ^T6K4 + To K3 K5 " "2^K3K4 + ^ 2 K 3 } T6 a 18. , 1 2 1 3s 385 *-12_4 x (- ^ 1 ^ + ^ K 5 ) + 3 ^ a . Thus, d . x . Fn(>/fx_). ~ * ( l - 8 l g n ( c ) ) + e - ™ c + n K < c > ( l + ^ + ^ ) 072701X2 n (1.4.18) , where c i s obtained from (1.4.6) and d g 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 F n„ .. On l e t t i n g p = c JnK^ and b ' = b r / i r (r = 1 , 2 , . . . ) , we have, by regrouping the factors i n the integrand of (1.4.4), CO r / rrr? (1.4.19) 1 " P n r b V r / = • * ( l - s i g n ( c ) ) + K(;c,n,x) f ' . n ( u | e r=3 CO S ;b ,{iu) r/;/H r" 2 ,v • r=3 - v: ". - » •-, But e = £ g T , ( i u ) ( / - J , where S r(y) ( r'- 0,1,2,...) are defined i n the obvious manner from equation (1.4.11). • • ' . ' / Define Q k(p) (k = 0,1,2,...) by Q ( p ) = J n{u)_ ( i u ) k du (1.4.20). -» p+iu Then, V p > = nTpJ + fi1^0)! ' N ( P ) ) (1.^.21), Q x(p) = 1 - p Q 0(p) ; (1.4.22), 19. and . Q k ( p ) (k = 2 , 3 , . . . ) s a t i s f y the recur rence formulae Q ^ p ) = "P ^ 2 k - l ( p ) ( k = 1 > 2 » ' " ) ( 1 . 4 . 2 3 ) , Q 2 k _ 1 ( p ) = ( - i ) k " 1 ( 2 k - 3 ) ( 2 k - 5 ) . . . ( 3 ) ( l ) - p Q 2 k _ 2 ( p ) . ( 1 . 4 . 2 4 ) . i Thus, f o r m a l l y , we o b t a i n , by i n t e r c h a n g i n g summation and i n t e g r a t i o n , a r e s u l t of the form - oo k 1 " P n V ^ r ) = ^ ( l - s i g n ( c ) ) + K ( c , n , x ) I h k ( p ) ( — i ( 1 . 4 . 2 5 ) , n k=0 Vn where h Q ( p ) = 0 . 0 ( p ) , h - ^ p ) = ^ a * _ 3 K 3 Q 3 ( p ) , h 2 ( P ) =^74 a * ' 4 K 4 Q 4 ( p ) + a * " 6 K 2 Q 6 ( p ) , ( 1 . 4 . 2 6 ) h 5 ( p ) = a * _ 5 K 5 Q 5 ( p ) + ^ a * - 7 K 3 K 4 Q 7 ( p ) 1 -*_Q "5 , . + i296 ° K 3 ^ ( p ) ' - M p ) = Y5o a * " 6 K 6 % ( p ) + (TT55 K 4 + T ! O k 3 K 5 ) o * " 8 q 8 ( p 1 *-10 v 2 I _ A / \ , 1 *-12 I_4_ , v + THZ8 0 K 3 K 4 Q l o i p ) + 3110^ 0 K 3 Q 1 2 ( p ) , and where the , Q k ( p ) (k = 0 , 1 , . . . , 1 3 ) are r e a d i l y obta ined u s i n g equat ions ( 1 . 4 . 2 1 ) , ( 1 . 4 . 2 2 ) , ( 1 . 4 . 2 3 ) and ( 1 . 4 . 2 4 ) . The approximat ion obta ined from ( 1 . 4 . 2 5 ) by d e l e t i n g a l l terms i n v o l v i n g h^ (k _> 5) w i l l be r e f e r r e d to as the sadd lepo in t 2 approximat ion . ! 20. 1 .5 Remarks1 This l a s t approximation w i l l be shown, by numerical means i n 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 . 1 8 ) , which f a i l s for c = 0 . A family of formal series expansions of . 1 - P can be obtained i n the manner of ( 1 . 4 . 2 5 ) by l e t t i n g c take any value i n (-c-^Cg) . In p a r t i c u l a r , c = 0 y i e l d s the Edgeworth series, which Cramer [ 3 ] proved to be asymptotic. Hence, i f the solution c of ( 1 . 4 . 6 ) i s 0 , the Edgeworth and saddlepoint 2 series must be i d e n t i c a l . However, as c moves away from 0 , the quality of the Edgeworth approxi- • • mation deteriorates. The explanation for t h i s l i e s i n the f a c t that c i s a saddlepoint, and, as i s well-known, through any saddlepoint there i s a path of steepest descent (see, f o r example, [ 6 ] ) : As Daniels [ 5 ] has argued, the path of inte- gration . [z ':. z =. c+iu,, < u .<<»} w i l l c l o s e l y approximate the path of ,'steepest, descent l o c a l l y in' the' region near the saddlepoint, from which the 'Only' asymptotic contribution.,to the i n t e g r a l comes. 21 . CHAPTER II THE SADDLEPOINT APPROXIMATIONS In this chapter we prove that the expansions given I i n ( 1.4 . 18) and ^ 1 . 4 . 2 5 ) are asymptotic. To do so we use Laplace 1s 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,, t h i s method i s r e a d i l y adaptable to other applications • involving:parameters'different from, n which tend to i n f i - nity.-•' • An example w i l l be- given.' i n Chapter 4. • 2 . 1 Asymptotic Expansions Returning to ( 1 . 4 . 8 ) , , for c ̂  0 , I ( x , n ) = ^ f e ° e - n x ( c + i u > + n n K ( c + i u ) _ d u _ ( 2 . 1 . 1 ) . -» c+iu i u - 1 7 Let g(u) = (1 + — ) " . Then g(u) i s a integrable over the i n t e r v a l (-R,R) fo r a l l R and i s equal to the convergent power series g(u) = 1 - AH + (IH)2 _ (IH)3 + . . . . {2.1.2) i n the i n t e r v a l [ - 6 ^ , 6 ] , where 6 ^ < c 22. As we s h a l l show subsequently, the asymptotically dominant contribution to the in t e g r a l i n (2.1.1) comes from that portion of i t which i s over [-n ,n~^~^]. In order that g(u) be adequately represented over t h i s region by th.4 1/3 f i r s t few terms of i t s series expansion, n must be large i n comparison to c~^" . Since i n the case of the saddle- point-1 approximation we assume g(u) i s so represented, i t tends to give unsatisfactory r e s u l t s when c i s near 0 and ,n . i s moderate1.,-,; The, saddlepoint 2 approximation leaves g . i n i t s o r i g i n a l form and tends to give good r e s u l t s even when / c ' is'small. ' However,, i t i s somewhat more complicated than the saddlepoint 1 approximation. ' : Let CO r h(u) = I a u , |u| < 6 , r=2 r h 1(u) = -ixu + K(c+iu) - K(c) , (2.1.3) where a r = K^ r^(c) i r / r J (r = 2,3,...) and 6 i s some pos i t i v e constant. . / Daniels [ 5 ] showed that i f the moment generating ' function M exists i n a region D = [z : -c^ < Re(z) < c 2 ] where D i s the largest such region, one r e a l root c of K^^(-z) - x = 0 exis t s , and i t s a t i s f i e s -c-^ < c < c 2 As M(z) i s analytic i n D , M(z) = M(c) + M^ l ) ( c ) ( z - c ) + M^ 2 )(c) ( z - c ) 2 + ... (2.1.4) 21 23. converges i n a c i r c l e with non-zero radius of convergence R Therefore, (2.1.4) converges uniformly on {z : |z-c| _< p} for some po s i t i v e p < R . Hence, -log M(z) = K(z) has a power series expansion about z = c which converges uniformly on| [z ; | z-c | _< 6 21 for some &2 > 0 . Thus h-L(u) has a power series expansion which i s uniformly convergent f o r i u j y>< &2 . \ that i s , h x(u) = h(u) l2 = a 2u + £ a u r,. ( |u| < 6 ) v ( 2 . 1 . 5 ) . r = 3 ' LEMMA 2, 1.1. Let 6 = min(6- L,6 2) . If 00 , I icp(t)i j' dt < (2.1.6(a)) for some r e a l j > 1 ,• then -6 and r- - nh,(u) j g(u) e l V du =.0(n- M). , nh, (u) [ g(u) e 1 du = 0(n" M) 6 ( 2 . 1 . 6 ) f o r each p o s i t i v e integer M PROOF. If the [X±] ,. ( i = 1,2,. . . ) , do not have a. l a t t i c e d i s t r i b u t i o n , 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 - o o ^ oo nh-.(u') some j _> 1 , say k . • Writing L N = " " T M - J g ( u ) e d u n we obtain !L N! < -TM . f n " k : . | M ( c + i u ) | k du - A.n M/p- n f o r some constant A > 0 . Thus lim |L I = 0 , since n^» P < 1. . r " 5 n n i ( u ) M S i m i l a r l y , J g(u) e du = 0(n ) LEMMA 2.1.2. There exists a p o s i t i v e constant p such that. Re{h(u)} _< -pu for a l l |u| _< 6̂  where 6̂  i s some, po s i t i v e constant. PROOF. E x p l i c i t l y , when |u| < 6 , h i s given by .h(u) = S K ^ ( c ) ( - l j r u 2 r + i E K< 2 r + 1\c) > l ) r u 2 r + 1 r=l ( 2 r ) J r=l" (2r+l).' Therefore, / Re{h(u)} = -K 2 + K̂  u 2 - Kg u" + . .. u where K i = K ^ 1 ^ ( c ) (i'=2,3,...) . Now, Re{h(u)}/u 2 i s continuous, and equals -Kg at u = 0 . Using the ter - minology and r e s u l t ( i i ) of Theorem 1.4.1, K ^ ^ ( t ) increases continuously from x = a to x = b . Hence, Kv ; ( c ) > 0 . 2 5 . Select € subject to the requirement Kg > e > 0 By the d e f i n i t i o n of continuity, there exists 6-̂  > 0 such that i f |u| _< 63 , * Re{h(u)}/u| _< -K p + e < 0 ^ 2 Thus,.; • Re(h(u)} _< -pu f o r a l l |u| _< 6^ and for some P > 0 ' . (Prom now on, l e t ' 6 = min( 6̂ , 6g, 6^) . v ... \ F i r s t consider the saddlepoint 1 approximation. We expand. g(u) exp(n S a u r) = g(u) exp^nu^ Y a.^1"" ) " r=3 r=3 double power series i n the two arguments nu y and u , convergent f o r a l l |u| < 6 . Denote th i s power series by • P(nu 5,u) = E r c. . ( n u Y J (2.1.7) i=0 j=0 1 J where the c. . ( i = 0,1,... ; j = 0,1,...) are independent of n. and u ' . In order to approximate P uniformly, by i t s p a r t i a l • "5 ' sums, we r e s t r i c t nu^ to some f i n i t e i n t e r v a l , say, |nu 5| _< 1 , that i s , |u| _< n ' 1 ^ = 6 n . . ' _3 We can assume n > 6 , so that &n < 6 . Then our region of integration f o r I(x,n) consists of f i v e i n - t e r v a l s : (-»,-6) , (-6,-6n) , (-6n>&n) > ( 6 n> 6) a n d ( 6' r o) • If 6 could be allowed to depend on n , we could take 6 = 6 n and thereby obtain only three sub-intervals of i n t e - gration and a s i m p l i f i c a t i o n of the proof. However, since 1 •26. 6 n - 0 ,' |M(c+iu)/M(c)| would not then be uniformly boun- ded by p < 1 for 6 <_ u < eo and a l l n , as i t must i f the r e s u l t of Lemma 2 . 1 . 1 i s to hold. LEMMA 2 . 1 . 3 . .Using the same notation as above, ; j " 6 n g(u) e n h< u> du + f g(u) e n h ( u ) du = 0(n" M) ( 2 . 1 . 8 ) - 6 c n f o r 'each p o s i t i v e integer M PROOF. For u e (6 ,*) , 2 . 2 pn(u - 6 n) > pn 6 n (u - 5 j > p(u - - 6 j ' (n > l ) n n' n • Then, 2 2 2 2 p n f i n ' r 5 ~ P n u ' p n 6 n f™ - P n u e j e du < e J e du 6 n 6 n • » -p(u-S') _ i < e n du = p . . . n 6 -pnu 2 y -pn 1/ 5 . Hence, J e du = .Ô e / J From th i s and Lemma 2 . 1 . ^n f6 g(u) en h(u> du + J " 5 n g(u) e n h ^ u ) du 6n :- " s < J :|:g(u)||e n R e h( u)|du + f"n | g ( u ) | | e n R e h ( u ) | d u n -6. < f e " p n u du + f n e " p n U du 6 n - 6 27. = 0(e p n . ) (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- p n u. 2 u M du = , 0 ^ e - * P n l / ° ; ( 2 . 1 . 1 0 ) 6 n PROOF. If u _> 6 n , we have uM e-|pnu 2 < UM e-|pu 2 < K (n > 1) for some constant K independent of n' . and so 2 u M = 0 ( e - n u ) = 0 ( e ^ p n u ) . Equation ( 2 . 1 . 1 0 ) follows by replacing p .with ^p i n 2 1/3 J e - p n u du = 0 ( e - p n ) It i s now clear why c i s chosen to be a saddle- point, that i s , a root of equation ( 1.4 . 6 ) . If this.were not the case, h(u) would include a l i n e a r term i n U , and P would become E T, c'-- (nu) u J . R e s t r i c t i n g nu to 1=0 j=0 ^ a f i n i t e i n t e r v a l i n order to approximate P uniformly by - 1 -1/3 i t s p a r t i a l sum leads to 6 n = n rather than n The proof of Lemma 2 . 1 . 3 w i l l f a i l since i n t h i s case 2 n 6^ 0 rather than » as n - co In the remaining i n t e r v a l , (""^n^n^ 3 P i s l28. approximated by it's p a r t i a l sums. For any p o s i t i v e integer A we write P (nu 5,u) = r • c.. ( m r 3 ) 1 uJ' . A i>0,j>0 1 J i+j<A LEMMA 2.1.4. If |u| < 6 n , P(nu 3,u) - P A(nu 3,u) = 0 [ ( n u 5 ) A + 1 ] + 0 [ u A + 1 ] (2.1.11), uniformly with respect to u and n (but not necessarily A), PROOF. Suppose an ar b i t r a r y power series, £ d z m y n , converges f o r |z| < R , |y| <S . , iri>0,n>0 11111 Since the terms of a convergent power series are bounded, dmn = ° ( R " m s " n ) • Then, i f |z| < R and |y| < S , E ^ dmn Z V = °( * i | | m | | | n ) m_>0,n>_0 m n m>0,n>0 K & m+n>A m'+n>A / = o( r (||| + |||)k) k=A+l K b = 0((||| + | | | ) A + 1 ) . Since, i n general, |a +•b|r <. 2 r _ 1 ( | a | r + |b|r) for a l l r > 1 , i t follows that £ , dmn ^'V 1 = 0( | z | A + 1 ) + . 0 ( |y| A + 1) m>0,n>0 m+n>A 29. Equation (2 .1 .11) i s an iriiraediate consequence. We now come to our main theorems i n which we s h a l l prove, with the aid of the preceding lemmas, that the ex- pansions given i n (1.4.18) and ( 1 . 4 . 2 5 ) are asymptotic. THEOREM 2 . 1 . 1 . Let d. = ( - a 9 ) " H I c 0 . m ( - a 9 ) m l ^ 2' m = Q m,2i-mv 2' x T f r n + i + i ) > (i= 0 , 1 , 2,. . . ) , where (2) a 2 ~ " K ( c ) / 2 * the t c m n } are defined i n equation ( 2 . 1 . 7 ) , and p. denotes the- gamma function, that i s , (2n).' . i 1 r(n+*) = nTi25""""" ( 2 . 1 . 1 2 ) . Then,, i f •, J M J < .» for some j _> 1 , CO • nh n (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, r e c a l l i n g that a 2 < 0 , / f,-8n. ' ,» i ^ n a ? u 2 . ! a 2 n 1 / / 5 { J . n + j . } P A(nu^,u)e ^ du = o(n Ae d • j (n-«) -eo 6 n (2.1.14) for any fix e d 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" na pu i|J g(u)e 1 d u " J _ w P A(nu 5,u)e 2 — 00 30. 2 dul •r̂ -'S nĥ(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 } P A(nu 3,u)e 2 du| 6 u 2 + |J n [ g ( u ) e n h ( u ^ - P A ( n u 3 , u ) e n a 2 U ]du| ~ 5 n *a n 1 / 3 - 0(n" M) + 0 ( e * 2 n A) 2 .oo na~u — CO + o(J. e 2 (|nu3|A+1 + |ulA+1)du) (n-~) (2.1.15), Now l e t us consider integrals of the type^ > r— -tx 1 - k e. x dx , where Re(t) > 0 ... For even k , sub- stituting'. tx = y ,•we obtain r e - t x 2 x2m. d x = t -m-| r ( m + J ) - = , - m - i ! ^ ^ ( : m = 0 , l , . . . ) mJ 2 (2.1.16). The folloxving estimate i s v a l i d for both odd and even k : 00 2 | |e- t x x k j d x = 0 ( [ R e ( t ) ] - * ( k 4 " 1 ) ) .(2.1.17) — oo using (2.1.17), we see that the l a s t term i n (2.1.15) i s 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 ( - | [ 5 m + k + i ] ) + o(n-* A- 1) + 0(n" M) ( n - ) , where i s 0 or 1 depending on whether i i s odd or even, respectively. As A and M are completely a r b i t r a r y , ! we obtain an asymptotic series » nh.,(u) » i . g(u)e du ~ E d.n * (n-») . -,M - c o . i=0 1 where the -d^ are. the c o e f f i c i e n t s computed i n Chapter 1 (1.4.17) and given, i n the statement of the theorem. J''; „•:•'*•' •'- • A, \ •'' nu^ E a u - • '.- .•' ••" . r=3 r » „j K(j) . • THEOREM 2.1.2. Let e = E • -i f E j=0 J' k=3j x c, ({a })u k , where a = K ( r ) ( c ) i r (r=3,4,...) , r i K(0) = 0 , K(j) •'= 00 ( j > 0) /'and - c k({a r}) (k=0,3,4,. .. ) are appropriately defined constants depending only on . (a r) (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 ( _ 2 a 2 ) r / 2 x n j _ k / 2 Qk(c,/C2lva^)/j.' ' ( n — ) (2.1.17) '32. ••is-, an^asymptotic expansion. PROOF. The proof i s almost i d e n t i c a l to the proof of Theorem 2.1.1 and w i l l only be outlined b r i e f l y here. . ' . n u 5 2, a u r ~ 3 In t h i s case, the factor e * . i n t h e integrand of (1.4.19) i s expanded as a power.series i n the 3 3 argument (nu^) . This series i s denoted by P'(mr) As i n the proof of Theorem 2.1.1, mr i s r e s t r i c t e d to the •interval |hu3| _< 1 , that i s , |u| _< rT1^ = -6 , i n order to approximate P' uniformly by i t s p a r t i a l sums P^ Lemma 2.1.3 can be applied without a l t e r a t i o n , while Corollary 2 2 2.1.3 i s obviously v a l i d with e " p n u • replacing e " p n u , c+iu since the l a t t e r . i s less i n modulus than the former everywhere. In the same manner as we obtained (2.1.11), we fi n d that, f o r lul < 6 , n (P' - P')(nu 3) = 0 ( [ n u 3 ] A + 1 ) A uniformly i n u and n . Then / 2 \ [, g(u)e 1 du = I p;(nu 3,u)e 2 -oo - a , C + X U + 0(n-«) + 0{f e^lnu'l ^ d u } , (2.1.18), Writing p;(nu 3) = E ( E a u r) J' j=0 J' r=3 33. 'A j K(j) k n = ^ i i _ £ c, ({a }) u , equation (2 .1 .18) . becomes 3=0 J' - k=3j- K ; • , c nh (u) A K(j) , f , - r n J ' _ k / 2 Qk(c,/T2naJ) j2lT/y. + 0 ( n _ M ) + 0(n"^ A" 1) (n-eo) ( 2 . 1 . 1 9 ) . Again, as M and A are e n t i r e l y a r b i t r a r y , the above, and hence ( 1.4 . 2 5 ) , i s an asymptotic series. This completes the proof of the asymptotic nature of the saddlepoint 1 and 2 approximations. We now discuss b r i e f l y a case not included i n Theorems 2.1.1 and 2.1.2, 2.2 The La t t i c e Case. When the [ X ^ , (i=l,2,...) , have a l a t t i c e d i s t r i b u t i o n , the preceding argument f a i l s ; the t a i l s of the i n t e g r a l (2 . 1 . 6 ) cannot be ignored, since for a d i s t r i b u t i o n having i t s mass concentrated at points h units apart, the c h a r a c t e r i s t i c function i s p e r i o d i c of period X = 2Tr/h , with |cp(X)| = 1 and |c?(s)| <,1 for 0 < s < X . Daniels [ 5 ] , i n h i s work involving the density func- t i o n , avoids t h i s complication because he i s dealing with densities instead of d i s t r i b u t i o n s . In that case, the path of integration stops at c + ITT , and no t a i l regions are present. Using t h i s 1 f a c t , an approximation to the d i s t r i b u - t i o n function could be obtained by numerically integrating the density function. F e l l e r [7] introduces the concept of a polygonal approximant F* to F n , where, more generally, G # i s the . convolution of G v/ith the uniform d i s t r i b u t i o n on (-h/2, h/2) and- h i s the span of G . He then shows that the f i r s t two terms of (1.2.2) approximate with an error of magnitude> o(n"^) . This means that at the l a t t i c e points of F n , the error i s o(n~ 2) when F n ( x ) i s replaced by |-[F n(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|-« i s necessary, a condition not met by l a t t i c e d i s t r i b u t i o n s and a considerable number of other d i s t r i b u t i o n s which have t h e i r v a r i a t i o n concentrated i n a set of Lebesgue measure zero. The order of magnitude of the error i n 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 i f a l l the moments of F are f i n i t e , i n the case of ! discrete d i s t r i b u t i o n s i t i s necessary to supplement the expansion (1.2.1) with discontinuous terms. ' However, although i t i s impossible to approximate such"discrete distribution.functions with continuous functions to an accuracy of within .one-half of th e i r maximum jump, l o c a l limit-theorems for approximating F n at i t s points of d i s - continuity exist (see Gnedenko-Kolmogorov [ 8 ] ) . We reproduce the' following proposition of Esseen (see [ 8 ] , p. 241) which i s analogous to the Edgeworth expansion (1.2.1) i n the absolutely continuous case. 35. Suppose the random variables X̂  ( 1 = 1 , 2 , . . . ) can only take on the values x = a + sh (s = 0 , +1, + 2 , . . . ) , o """" where h i s the maximum span of the d i s t r i b u t i o n P .. The random variable •n n oM k=l K can only take on the values yns = h ( S " nP)/(°^ H) > where p = r, i p i and p^ = P(Xk = a+ih) . Let l = - e o P n(s) = P(Yn = y ^ ) . THEOREM 2 . 2 . 1 . (Esseen). If the i d e n t i c a l l y d i s - 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 ( Y n s ) ) . + 0 ( n " ( k _ l ) / 2 ) '• ( n — ) . Here, R l ( n ( y n s ) } = " 37 * 0 ) ( y ) > D , , u Xk (k),• . 10 , 2 ( 6 ) , x R 2 ( n ( y ) ) = TI n (y) + sr x 3 n ( y ) ' and the R^(n(y)) ( i > 2) are obtained i n 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 F n at the points of discontinuity y now only requires a summation procedure, F (y ) = F (y , + 0) n u n s ; n V J n , s - l ; = S P (r) r<s / 37. CHAPTER 3 y. ' •;• .'. . "' COMPUTATIONS . . : ;,. ̂ T o judge the quality . o f t h e saddlepoint'approximations in the case, of small • n several test cases were considered. Numerical re s u l t s were obtained i n 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 f o r values of n between 1 and 40 inclu s i v e . For bre- v i t y , only a few representative r e s u l t s f o r each d i s t r i b u t i o n considered are depicted. * . It w i l l be noted that whereas the saddlepoint 2 expansion gives uniformly better r e s u l t s than the other three approximations, the Edgeworth series, (1.2.2), i s quite good when c i s close to 0 , as we would expect on the basis of the discussion i n section 1.5. However, when x assumes values i n the extremities of i t s range, the saddlepoint method gives s u b s t a n t i a l l y better r e s u l t s . 3-1 Remarks on the Tables. When e = 0 , the saddlepoint 1 expansion does not e x i s t , and for programming purposes, the- Edgeworth ap- proximation i s printed i n i t s place. When F (x). i s nearly 1, exponents i n the calc u l a - tion of the Cramer formula are excessively large f o r the com- puter, and the value F (x) = 1 i s 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 f a c i - l i t a t e a comparison of the rates of apparent convergence of the various series. The r e s u l t s are printed i n exponential format, and a series of d i g i t s , say O.n-̂ . ..n^ , followed by "D +. m" +m represents the number O.n-̂ . ..n^ x 10— , The l e t t e r E occasionally replaces the l e t t e r D i n t h i s format. In order to observe the e f f e c t of the location of the saddlepoint on the various approximations, the value of c and the accuracy to which i t i s calculated i s given. Hence, "saddlepoint = + or - 5" means c e (c^-6,c^+fi) When i t i s available from e x i s t i n g tables, the correct value of P n ( x ) l S given f o r comparison. Prom these cases i t appears that the l a s t entry for the saddle- j • point 2 approximation i n each case i s accurate at least in the d i g i t s where it . and the next to l a s t entry agree. In the .remaining cases,, judgement on the quality of the various approximations must be w i t h h e l d 7 u n t i l exact values become /available. If .the l a s t saddlepoint 2 entry i s accurate to the extent just.described, as seems l i k e l y to be the case, , - an examination of the tables indicates that t h i s method of approximation gives r e s u l t s of the same comparatively good quality as i t did i n the e a r l i e r cases f o r which exact values of F (x) are known. 39. 3. 2 Chi Random Variables. Let X± = |Y il where Y± ( i = 1,2,...) are independent, standard normal variables. The density function of i s given by ET 2/P ! f (x) = VTT e" x 7 , x 2 0 0 , x < 0 , and the moment generating function i s M(t) = 26* / 2 N(t) , 1 v 2/2 where ' N(t) = J e y / dy — eo The cumulants are given by Rubin and Zidek [13] as ^1 = a i Z 0-79788 45608 03 a 2 = a 2 = 1-a 2 ~ 0.36338 02276 32 a 3 a 1 ( a 2 - a 2 ) ~ 0 . 2 1 8 0 1 3 6 l 4 l 45 a 4 = 2 a 2 ( 2 - 3 a 2 ) ~ 0.11477 06820 54 (3-20a 2+24a^) ~ - 0.00443 76884 6262 The saddlepoint c i s 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 i n [ 13] ; This entailed the evaluation of M(z), and hence, of N(z) , for complex values of i t s argument. Since use of i t s Taylor ex- pansion r e s u l t s i n uncontrollable round-off error, an ac- • curate method using continued f r a c t i o n s was employed. The computation of . Q Q(p) (see equation ( 1 . 4 . 2 1 ) ) also requires W(p) , and for t h i s reason, a detailed account of the me- thod devised i n [13] i s given i n Appendix A. The derivatives of the cumulant generating function evaluated at c are K-̂  = x/n = x _ .. _2 K 2 = ex + 1 - x K^ = c 2x + c ( l - 3 x 2 ) - x ( l - 2 x 2 ) K 4 = c \ + c 2 ( l - 7 x 2 ) - cx(5-12x 2)' + x 2(4-6x 2) K 5 = c 4x+c 3(l-153c 2) - c 2x(l6-50x 2) - c ( 3 - 3 5 x 2 + 6 d x 4 ) + x ( 3 - 2 0 x 2 + 243c4) K 6 = c^x + c^(l-31x 2) c % ( 4 2 - l 8 0 x 2 ) - c 2(13-191x 2+3903c i |')'+ cx ( 4 l - 2 7 0 x 2 + 3 6 0 x l 1 ) - x 2(28-120x 2+120>c 4) . n From now on, F (x) w i l l denote P( £ X. < x) . Then, f o r n  f i = l 1 ~~ the saddlepoint 2 method, the f i r s t 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 3 8 ' 0 1 + 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 . 0 0 0 0 0 = -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 0 0 . +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 . 6 2 7 4 E - 0 4 . 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 . C R A M E R 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- F n ( x ) i s . , >> F n ( x ) sign(c)) - e - x c + n K ( c ) Q Q ( p ) A / ^ , where Q Q ( p ) i s given by (1.4.21). • " ' •1 V . The res u l t s xve obtained i n t h i s case are' l i s t e d i n Tables 1(a), (b)." The exact values of F n ( x ) 'given are those computed by Rubin and Zidek [13]. 3-3 The Exponential P r o b a b i l i t y Law. The p r o b a b i l i t y density function of the exponential d i s t r i b u t i o n i s given by , x > 0 ,• otherwise where X > 0 . The moment generating function i s 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 y i e l d s f(x ) = Xe 0 -Xx c = X - (x/n) 4 4 . and hence, K^ r^(c) = r< X (x/n) r (r = 1 , 2 , . . . ) . D i f f i c u l t i e s are encountered i n 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 . 1 2 1 3 „, X(z) = j - ^ Z + ^ Z - - - g Z + . . . , which does not converge for |z| > 1 . Thus, i f x > 20 and "" n = 1 0 , vj/Jn = (x-nfi )/(no) > 1 , and x(w/7n) cannot be' evaluated using ( 1 . 3 - 4 ) . We overcame t h i s d i f f i c u l t y by i n v e r t i n g d i r e c t l y the equation ( 1 . 3 - 3 ) . The form of the expression f o r c indicates that i t can be considerably d i f f e r e n t from 0 for moderate values of x . Hence, the Edgeworth series often yielded inaccurate r e s u l t s . For example, when x = 4 and n = 15 i t i s incor- rect i n the f i r s t .-significant figure. Results f o r t h i s case are l i s t e d i n Tables 1 1(a), (b). , i 4_5 EXPONENT IAL RANDOM VARIABLE ' M E A N = 1 VARIA N G E=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 . 9 3 • E - 0 3 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 i a b _ l e _ j n i ( . 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. 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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 U J 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 C J c- Co o CJ,' *ovo o L O LP. o •-C f̂- 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 • x r > 1 o o cc LO. c 6 .' Q: NT' d r H 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- 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 — • i i <l" — H- X ii. ex. o C J 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 P J C : rv cv CV cv cv P J P-J rv P-J rv ' Jr-" o o o C~' o . Jf- > £ o . J - C J 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. U J rv rv rv c-J ; LJ rv r-vi rv rv CV! ' UJ --o LL; CJN CJ- 0 S 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 - 0 1 . 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 P r o b a b i l i t y Law. : This case i s considered because extensive and . highly accurate tables are available. The r e s u l t s obtained indicate the high accuracy possible with the saddlepoint 2 approximation. In a l l cases considered, the f i r s t term i n ( 1 . 4 . 2 5 ) gave an answer which i s correct to every figure tabulated. However, as the r e s u l t s given i n Tables I l l ( a ) and (b) indicate,•' the Edgeworth and Cramer methods generally incorrect i n the l a s t two or three figures. 3.5 The Non-Central ChirSquare P r o b a b i l i t y Law. The d i s t r i b u t i o n function of the [X i) ( i = 1 , 2 , . . . ) i s given by F(x|v,\) = T. e ~ ; / 2 (X / 2 ) J F j x , y + 2 j ) , j=0 j.< C where X _> 0 i s termed the non-centrality parameter, v is,the number of degrees of freedom, and F fx,k) = [ 2 | k P ( i k ) ] ' 1 f X t | k _ 1 e - ^ dt (0 < x < .) u o .' ~ i s the central chi-square d i s t r i b u t i o n v/ith n degrees of freedom. The c h a r a c t e r i s t i c function of X^ i s cp(t) = e x p [ X i t / ( l - 2 i t ) ] ( l - 2 i t ) _ V / / 2 Using equation ( 1 . 4 . 6 ) , we r e a d i l y f i n d that c, the saddlepoint, i s given by 50. •- 2 Also, the derivatives of the cumulant generating function K evaluated at t = c are given by K ^ \ ( c ) - = ( l - 2 c ) " J ' 2 J ' - 1 ( j - l ) j [ v + \ J 7 ( 1 - 2 C ) ] ' . As a f i r s t approximation to F n ( x ) we can write, using (1.4.25), F n ( * ) - + sign(c)) + e - x c + n K ( c ) + p / 2 [ | ( l + sign(c)) - N(p)] , where p - cvnK^ 2^(c) . Numerical r e s u l t s are tabulated i n 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 N T 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 S A D O L E P G I N T 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 P r o b a b i l i t y Law. The p r o b a b i l i t y density function and moment genera- t i n g function, respectively, of a random variable d i s t r i b u t e d uniformly over th e . i n t e r v a l (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 s i m p l i c i t y , we consider the case a = -b In general, the cumulants, i f they ex i s t , may be expressed i n terms of the central moments {û } as a± = U i , i = 1,2,3 . a5 = U5- " 1 0 M M 3 + ^ l ^ 2 3 a6 = ^6 _ 15wy i 2 + lQ4- 5(W 5+6u 2u 1+2y 1) + 30,̂ 2 For the uniform probability- law, as i s e a s i l y shown, u i •"=:::• 0 i = 1,3,5, bV(i+l) i = 2 , 4 , 6 , 55, Thus, •a g = = a 6 = 0 ' b 2 / 3 2 15 b 5 b 6 TH b i = -1,3,5, Equation (1.4.6) becomes I', , / cb , -cb\// cb -cb N , — + b (e + e )/(e - e ) = x/n To obtain numerical results, for t h i s case, the .last equation v/as solved numerically f o r c with i n i t i a l i t e r a t e = x/n , and successive i t e r a t e s obtained by the Newton technique. Note. that since | K ^ ^ ( t ) | < b , a saddlepoint exists only i f |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 - 4 b 2/u 2 (t) = - 2 t ~ 3 + 8 b 5v/u 3 (t) (t) (t) 6 t " 4 + 8 b 4 ( l - 3 v 2 / u 2 ) / u 2 - 2 4 t - 5 - 3 2 b 5(2 - 3 v 2/u 2)v/u 3 1 2 0 t - 6 - 32b 6 (2-15v 2/u 2 +15v 4/u 4)/u 2 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 . 1 0 7 5 4 2 5 7 3 E - 1 2 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 5 4 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 . 8 5 7 5 6 4 8 3 8 E - 0 1 ...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 . 7 5 2 5 3 1 5 8 2 E 0 0 15 EDGEWORTH - . CRAMER "SADDLEPOINTI SADDLEPOINT 2 0 .7532185970 0 0 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 F J . X ) = . .0. 9 9 7 . O 6 I 7 4 9 . . 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 F n ( x ) i n d i c a - tes that the. saddlepoint 2 series again y i e l d s the most accurate r e s u l t s (see Tables V (a) and (b)). "\'3.7 Remarks >. In addit i o n a l test cases (which for the sake of •.brevity are omitted) involving random variables from sections : . 3 . 2 to. 3 . 6 , the"results obtained were q u a l i t a t i v e l y the same as those reported. Although for the reasons, cited i n section 2 . 2 these methods of approximation cannot be t h e o r e t i c a l l y j u s t i f i e d i n the l a t t i c e case, discrete random variables d i s t r i b u t e d according to the Poisson p r o b a b i l i t y law were treated. Predictably, the res u l t s were e r r a t i c and usually inaccurate, but when the argument x was a point of discon- t i n u i t y of F n , the saddlepoint 2 series yielded r e s u l t s ' which were accurate to two s i g n i f i c a n t figures i n almost a l l cases. Only the Edgeworth expansion, when c was.close to . 0 , yielded r e s u l t s of similar quality. 59- CHAPTER 4 OTHER APPLICATIONS. 4.1 The Non-Central Chi-Square Distribution. The form of the c h a r a c t e r i s t i c function of the non- central chi-square p r o b a b i l i t y law suggests that an alterna- t i v e approximation to the d i s t r i b u t i o n function of the n- f o l d convolution of t h i s lav; may be obtained by expanding the integrand i n (1.4..4) i n powers of \ , where \ i s the non-centrality parameter. The objective of t h i s 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 d i s t r i b u t e d random variables, each of whose d i s t r i b u - t i o n has non-centrality parameter X . The moment genera- ting;.:function of X., ( i = 1,2,...) i s . - V M ( t ) = exp[Xt/(l-2t)] ( l - 2 t ) " v / 2 ! where v = number of degrees of freedom and X = non-centrality parameter. Equation (1.4.4) becomes • 1 - P n(Xx) = | ( l - s i g n ( c ) ) +~f" [ l - 2 ( c + i u ) ] " V n / 2 60. x exp[-Xx(c+iu) + nX(c+iu)/(l-2(c+iu)"•') ]_du_ (4.1.1). c+iu The saddlepoint c i s the root'of the equation Thus, c = $(1 - Vx) . If we write g(z) = nz/(l-2z) , and proceed formally as i n Chapter 1, the i n t e g r a l i n equation (4.1.1) becomes exp[-Xxc + xg(c)] f [ l - 2 ( c + i u ) ] " n v / 2 e x p [ - x ( - g ( 2 ) ( c ) u 2 / 2 — 08 CD + E g< r>(c) ( l » ) r / r J ) ] ^ H r=3 This can be expanded as a series i n powers of X , the asy- mptoticity of which can be demonstrated i n a proof very similar to that of Theorem .2.1.2. This expansion, up to the f i r s t ! five.terms, i s • . .w:.'' -2Xxc 2 F n(>x) = |(1 + sign(c)) -/-!=- t h 0 Q 0 ( p ) + X'' i(h 1Q 1(o) + h Qb 3Q 3(p)) + X" 1(h 2Q 2(p) ' + (h xb^ + h Qb 4) Q 4 ( 0 ) + *h 0b|Q 6(p)) + X"' 5 /?(h 3Q 5(p) . + ( h 2 b 3 + h l b 4 + h 0 b 5 ) % ( p ) + ( * h i b 3 •+ h o b 3 b 4 ) Q 7 ^ p ) . +'T5 hG b3 Q9 (' j ) ) ^ x " 2 ( h 4 Q 4 ( p ) . + ( h3 b3 + h2 b4 + h i b 5 + h o b 6 ) Q 6 ( p ) + (* h2 b3 + h i b 3 b 4 + * ho b4 + h Q b 3 b 5 ) Q 8 ( p ) ;  + (ihl°l + i hoblHtelo(*\ + ^ T h o b 3 Q 1 2 ( p ) ) ^ > (4-1.2) 61. where 0 = c V X g ^ ( c ) . Q i ( p ) ( i = 0 , 1 , . . . , 1 2 ) are defined by equations ( 1 . 4 . 2 1 ) , ' ( 1 . 4 . 2 2 ) , (1.4.23) and (1.4.24), g ( 1 ) ( c ) x i b i = rm i = * ( n x ) s ^ ( i = 3, . . . , 6 ) ,. and . 1 i . ' V g ^ ( c ) 1 h. 0 ( i = 0,1,2,... ) , M denoting'the i . ' V g ^ ^ c ) 1 function M f t ) = ( l - 2 t ) - n v / 2 c . For the sake of s i m p l i c i t y , the case y = 1 was considered. Numerical re s u l t s given below in Table VI i n d i - cate that for moderate values of n and \ , the expansions are nearly equivalent i n accuracy and speed of "convergence". As expected, our e a r l i e r approximation is.superior where n i s large. But even i n "extreme cases, such as when n = 1 and X = 1000, the improvement achieved by using the new approximation i s very s l i g h t . 62. Table VI A • COMPARISON OF APPROXIMATIONS ( 1 . 4 . 25 ) AND (4 . 1. 2 )• TO Fp_( \x) X n x EQUATION ( 1 . 4 . 2 5 ) EQUATION ( 4 . 1 . 2 ) 100 15 15 100 1000 8 98 . 4 2 3 3 7 4 8 7 9 . 5 0 0 0 0 0 0 0 0 0 .428272116 . 4 2 7 8 9 5 4 8 2 .428277904 . 4 2 7 8 9 5 4 8 2 .428278337 . 4 2 8 2 8 0 0 4 0 .428278338 . 4 2 8 2 8 0 0 4 0 .138175666 . 1 4 1 8 0 9 3 2 2 .. 145448561 . 145367769 . 145528761 . 145535852 . 1 4 5 5 4 5 8 0 1 . 145545588 . 1 4 5 5 4 6 4 1 8 . 1 4 5 5 4 6 2 1 5 .369745696 . 373832713 .375295334 . 3 7 5 2 9 9 4 7 2 . 3 7 5 3 0 8 0 6 1 , 3 7 5 3 0 8 8 2 7 .375308876 . 3 7 5 3 0 8 8 9 6 . 3 7 5 3 0 8 8 8 0 . 3 7 5 3 0 8 8 9 6 4.2 The Doubly Non-Central F-Distribution. 2 2 • Let and x2 ' o e ^ w o independent non-central chi-square raiidom variables with degrees of freedom f ^ and fg and non-centrality parameters X^ and Xg , respectively. The d i s t r i b u t i o n of Xp = ^ i ^ i ^ s c a H e c * the doubly non- ~2 X 2 / f 2 central P-distribution. It occurs i n the analysis of v a r i - ance and i s used i n engineering problems where i t gives the p r o b a b i l i t y of error i n certain communications systems. No simple formula for evaluating the p r o b a b i l i t y i n t e g r a l , F , of Xp i s available. Tiku [ 1 5 ] developed several series expansions f o r F which y i e l d s a t i s f a c t o r y approximations when ( i ) X j i s large and \g i s small , ( i i ) >, 1 i s small, and \g i s large , ( i i i ) both X-, and Xg are large. • - . . In t h i s section we obtain an alternative to Tiku's approximation f o r the case when f ^ and fg are large and . X-| and Xg are moderate. It i s derived by means of the saddlepoint method and i s , i n part, intended to demonstrate the v e r s a t i l i t y of t h i s method. Gurland [9] shows that. i f X-, and Xg are two independent random variables v/ith c h a r a c t e r i s t i c functions and cog > respectively, then the r a t i o X-,/Xg has a d i s t r i b u t i o n function G which s a t i s f i e s -l -e T G(x) + G(x-O) '= 1 - — r - lim lim (f + \ V, (t W - t x ) d 1 n T-» e-»0 -T *V 1 ' d ~~i (4.2 . 1 ) . - f /2 Putting co,(t) = e x p [ x , i t / ( i - 2 i t ) ] ( l - 2 i t ) J (j = 1,2) L> J we obtain from equation (4.2.1), since F i s continuous, 2 P ( f a x / f 1 ) - 1 - K lim lim {J"% / } e x p r V L - V " T 0 e J ^ l T T 2 i t T T 2 x T - f n l o g ( l - 2 i t ) - f 2 l o g ( l + 2 x i t ) ] ~ (4.2. _- _^ If we. suppose f ^ » while X-^Xo a n d- -g a r e 'ixed, the appropriate saddlepoint equation i s 0 This equation has no f i n i t e solution, c . A similar prob lem occurs when we t r y to expand i n terms of -2'^1 ° r If the saddlepoint method, i s to be useful, there fore, we must suppose that f ^ -* <= , fg -+ *> and that X~̂ and Xg are f i x e d parameters. Then c is'the solution of i y - i log(.l-2z) + - | log(l+2xz)] = 0 , which y i e l d s c = 2 xT. p ~ f ^ . Now, - < c < ; thus, formally at l e a s t , we can proceed as i n section (1.4) and expand the functions i n the integrand of (4.2.2) about c i n convergent power series. Let f ( z ) = \ z/(l-2z) - X pxz/(l+2xz) , • * 5 . g(z) = - tf± l o g ( l - 2 z ) - | f 2 log(l+ 2 x z ) * r ( 2 ) ' u * ^ x ( f r f 2 > , 1 1 Then, equation ( 4 . 2 . 2 ) can be'"rewritten' 1 ffn\ ( \ra en f( J'^(c) ? ( x f 2 / f 1 ) = i(l+sign(c))-- ei{c)+^c)j exp[ s — 7 — - ( i u ) J ] x exp[ r — r M l i u ) J exp[ g—JTTTTJ ( 4 . 2 . 3 ) r=3 Define constants a. ( j = 0 , 1 , 2 , . . . ) by *> • (r) / \ . r ro - j e x p [ v SLIM ( ! { ) ] . j a r=l a j = 0 J a and l e t bj = g ( ^ ( c ) / j < , ( j = 3 , 4 , . . . ) , and p = c Proceeding as In the derivation of equation ( 1 . 4 . 2 5 ) , equation ( 4 . 2 . 3 ) becomes, P ( x f p / f 1 ) = *(l+sign(c)) 1 - e f ( c ) + g ( c ) T. c j f ^ f ^ ) ~ X ./2ir J = 0 . J ~ ( 4 . 2 . 4 ) . Each c .(f ,f,_,) ( j = 0 , 1 , . . . ) represents a term of the order (f^ ^ ) • The f i r s t few terms i n the series of equation ( 4 . 2 . 4 ) are 66. c o ( f r f 2 ) = QQ(p) c 1 ( f 1 , f 2 ) = ' f l Q x (p) + ^ | Q 5 ( P ) i ' h 2 c 2 ( f 1 , f 2 ) = f | Q 2 ( p ) + ^ ( b 4 + a 1 b 5 ) Q 4 ( p ) + ^ _ ^ Q 6 (o) c 5 ( f 1 S f 2 ) = f 3 Q 3 (o ) + • ^ ( b 5 + a 1 b 4 + a 2 b 5 ) Q 5 ( p ) + i , f - T ( b 5 b 4 a a • . a c i | ( f 1 , f 2 ) = *4 Q 4 (p) + ^ ( b 6 + a 1 b 5 + a 2 b 4 + a 3 b - 5 ) Q 6 ( p ) a a ^ R 5 ( | b 2 + b ; 5 b 5+a 1 b 3 b 4+ia 2 b|)Q 8 ( p ) + - l ^ ^ b ^ . Here, Q̂ (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 X 2 are small, say X^ < 1 . The expansion is not, of course, uniform in \^ and X 2 , and is usally accurate to no more than 2 significant figures for larger . X̂  •/ When f = f 2 = f , the above approximation can be simplified considerably. The saddlepoint c. becomes -^(1 - -i) . If we letg(z) = X 1z/(l-2z)-X 2xz/(l+2xz) and h(z) = -\ log(l-2z) - \ log(l+2xz) , then F(x) = -|(l+sign(c)) - 1 e g ( c ) + f h ( 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 Q u2511567 . 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 c o = ' . c 1 = a1Q1(P") + b 5 Q 5(p) c 2 = a 2Q 2(p) '•+ (b^+a-jb^Q^p) + ^ ^ ( p ) 2 s c 5 = a 3Q 5(p) + (b 5+a 1b 1 |+a 2b 5)Q 5(p) + (fca-jb^+b^) : x G^(p) + ^b|<^(p) c4 = a 4Q 4(p) + (b 6+a 1b 5+a 2b J ++a 5b 5 )Q 6(p) + (b^+a-jbjb^) + t a 2 b | + ^ ) Q Q ( 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 ^ f c ) ( i = 3,4,... ) , and in i W h ^ ( c ) : a^ ( i =0,1,...)' i s defined by the equation co (r) / \ , iv v 1 ca ex P[ i g y ^ ( r r ^ - ) ] . 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 f i of the same order of magnitude. .: . In the case when \^ = Xg = X , a more suitable approximation is achieved for larger. X • say X > 3 ,: i f . - . ^ : . - ^ , ; . ; , - . . r . T a b l e V I I I . W ; 6a. . APPROXIMATIONS TO F - R AT In ' N l =NO.OF DEGREES OF FREFDOM IN NUMERATOR N2 =NO.OF DEGREES OF FREFDOM IN DENOMINATOR N . C . P . ( 1 ) = N O N C E N T R A L I T Y P A R A M E T E R . I N NUMERATOR N . C . P . ( 2 ) = N O N CENTRAL ITY PARAMETER IN DENOMINATOR N l N2 N „C . P . ( 11 N . C . P . ( 2 ) X F - R A T T O 0 . 3 7 0 8 7 6 6 .0....3.6.0.L6.5.2, 0 . 3 6 2 7 3 6 5 0 . 3 6 3 4 2 4 8 0 . 3634509 35 35 5 . 0 0 0 0 1 .0000 0 . 7 5 0 0 0 . 1 4 9 0 0 8 2 , .„.._ Q .1.26J945Z 0 . 1199288 0 . 1236284 ' * . _ 0 . 1245618 0 . 8 1 5 4 4 8 8 0... .8.1.5 4 4.8.8. 0 .80961.26 0 . 8 0 9 6 1 2 6 0 . 8 0 9 6 1 2 5 10 10 0 . 0 0 0 0 . 1 .0000 2 . 5 0 0 0 0 . 9 3 5 7 4 0 5 ...... L _ ... ; _ „ .0 . 93.8AU\3. . . . 0 . 9 3 7 6 1 9 4 0 . 9 3 7 4 2 0 7 • 0 . 9 3 7 3 0 1 3 10 10 1 .0000 1 . 0 0 0 0 ' 0 . 7 5 0 0 . • 0 . 3 2 5 3 8 3 7 „_„„: : " " • • • ' .0 ..33 500.03.. 0 . 3 2 3 8 592 . • •' .' V "• • 0 . 3 2 7 1 4 1 8 • '0 .3281402 9 9 0 . 5 0 0 0 0 . 2 5 0 0 ' 0 . 6 0 0 0 0 . 2 1 7 6 4 9 9 , w . *L- —: " • 0 . ? 163377' •< ' 0 . 2 1 7 2 7 2 9 ' . . \ " 0 . 2 1 7 3 9 7 0 • • ' •• ' - ' 0 . 2 1 7 6 0 4 9 15 15 1 . 8 0 0 0 . 1 .2000 ,1 .4000 . 0 . 7 2 8 1 9 7 0 _*_ _0L . 7.0A8.SL5 7„ 0 . 7 1 3 9 9 0 9 ' ' 0 . 7 1 8 52 32 0 . 7 1 7 0 7 7 5 0 . 3 2 2 3 0 1 3 __.Q...4.2 2 10.6.9 : __. 0 . 3 9 0 9 0 0 0 0 . 3 6 6 3 8 0 7 0 . 3 7 2 4 2 2 4 7 7 0 . 2 5 0 0 C . 0 0 0 0 . 0 . 7 8 7 5 7 7 • 0 . 0 0 0 0 0 . 0 0 0 0 2.0000 i 40 40 0.2.500 ' " 6 . 2 5 0 0 0 . 7 8 7 5 70. we expand the i n t e g r a l i n powers of• \ * . ,,' 'The appropriate saddlepoint equation now i s ' ' d r z • xz n _ ' This equation has root c = (-2L/x + x + l ) / ( (2,/x)(x-l))' : I t i s seen that - |— < c < -| , • and thus, the functions i n the integrand may be expanded about c Again, the approximation i s not uniform i n the remaining parameters, f ^ and f g , hut f o r moderate values of f , say f ^ < 15 > f a i r l y accurate r e s u l t s ( 3 or more s i g n i f i c a n t figures) seem to be obtained. These and others are tabulated i n Table IX. 4 . 3 - Remarks. The r e s u l t s of t h i s chapter suggest that the saddle- point method can be e f f e c t i v e l y applied to an i n t e g r a l of the ' form [ g(z) e*" h( z^dz , provided that the equation .• - i . e . h ^ ^ ( z ) = 0 has a f i n i t e , r e a l solution c . If i t does, ..•.v»'- ' f 0 0 , ^ h i ( u ) and the i n t e g r a l can be put i n the, form J g^(u) e du , where and . h^ s a t i s f y the conditions of Theorem 2 . 1 . 1 , :.' t h i s method w i l l - y i e l d an asymptotic expansion i n powers :of '• X~'̂  I f a s i n the example considered i n section 4 . 2 , ••'.the problem has other parameters i n addition to X , the expansion need not be uniform with respect to them. 71. APPENDIX A l . •' Computing N(z) . . In t h i s appendix i s presented a method of e v a l u a t i n g N(z) f o r complex values of z which was devised by Rubin and Zidek [13]. I t uses continued f r a c t i o n a l expansions f o r N and thereby avoids the u n c o n t r o l l a b l e round-off e r r o r s which accrue i n usi n g the Taylor's expansion. Their method i n v o l v e s the complex form of Shenton's [14] continued f r a c t i o n f o r small values of |z| and Laplace•s continued f r a c t i o n (see K e n d a l l and St u a r t [11], p. 138) otherwise. Since N(z) = 1 - N(-z) , we can without l o s s of g e n e r a l i t y assume Re(z)_> 0 . W r i t i n g a l b l +" a 2 ' = • f i _ a 2 • a 3 . ... b,+ b 0+ !>,+• : b 2 + a^ / 1 . d > b^ + ... ( A l . 1), we o b t a i n , u s i n g the Shenton f r a c t i o n , N ( 2 ) : ; = : l ; + n ( z ) v f - f . ^ ! r £ £ . . . ) , „,(,).> 0 (A1.2) and , u s i n g L a p l a c e ' s f r a c t i o n , N(z) = 1 - , . n ( z ) ( | - h h h , Re (z ) > 0 (A1.3). -2 1 R e w r i t i n g e q u a t i o n ( A l . 2 ) , w i t h t = z , (A1.4). We s h a l l c a l l | _ ^ , : „ ( g n - l)/[(4n-?)(to - l ) 1 • ( n = 1 ) 2 ) . . . ) ( A 1 . 5 ) the 2 n t h approximant t o the c o n t i n u e d f r a c t i o n i n ( A 1 . 4 ) . The remainder s a t i s f i e s ' 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 c o n t i n u e d f r a c t i o n on the r i g h t s i d e o f (A1.6) r e p r e s e n t s the f u n c t i o n u ( t ) which s a t i s f i e s the e q u a t i o n u = a n [ 1 ~ a n ( t + u ) " 1 ] - 1 (A1.7), t h a t i s , u(t) = ( a n - t/2) + [ ( t / 2 ) 2 + 73. (A1 . 8 ) , where = 1/8 n Let R n ( t ) = Re{(t / 2 ) 2 + a 2} I n ( t ) = Im((t/2) 2 + a 2} (A1.9) R n ' ( t ) = t * ( R n ( t ) + CR^(t) +.I^.(t)]*)]* ( n = l i 2 , . . Then (see Ahlfors [2], p.3) ± [ R ^ t ) + | I n ( t ) / R ; ( t ) ] (R^(t)+0) ' 0 otherwise (A1.10);-. The square root i n ( A l . l o ) has branch points at. + 2 a n i , and theafunction obtained by choosing either sign i s a branch of* the square root. Rather than f i x the sign, we take sign[Re(tj /][R;(t) + | I n ( t ) / R ^ ( t ) ] , Re(t)*0,R£(t)=}=0 k^t) + | I n ( 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 f r a c t i o n . 74. Using (A1.8) we obtain, as an asymptotic approxima- t i o n to the continued f r a c t i o n 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 s a t i s f a c t o r y r e s u l t s when Re(z) = 0 J. Sim i l a r l y , we obtain an approximation to the con- tinued f r a c t i o n of (A1.3), fe ^ fe ' : R e ( z ) > ° ' <» -a.5--'> z+ (z +4n) (A1.13). One addit i o n a l modification i s introduced to further improve these approximations. If H n ^ z ) = J ~ ( n - l ) i exp[-|(t+z) 2]dt , Re(z)-> 0 , (n =1,2,...) , (A1.14) and H 0(z) - e - 2 / 2 then H n(z) = (n+1) H n + 2 ( z ) + z H n + 1 ( z ) - , (n = 0,1,...) (A1.15). Le t t i n g = H n_ 1/H n y i e l d s 75. Q j z ) = z + n/Q n + 1 ( z ) , (n = 1,2*.'./) (Al. 16) . Hence,-- Q-^z) > n(.z)/(l-N(z)) = z + 1/Q 2(z) 1 2 _ ( n - l l ^ ^ = 2 , 3 , . . . ) (A1.17), = z + z+ z+ We- now replace 4n by a^ i n (Al. 1 3 ) , where the {a^} , (n = 2,2,...) , are chosen so that the approximation ^(zWz +a^) to Q^ 2) ' ̂ s exact at z = 0 , that i s , a; = sr2(^+i)/r2(|) , ( n - 2 , 5 , . . . ) , ( A I . I S ) where P denotes the Gamma function. 0 Let R n ( z ) and I n ( z ) denote Re(z +a^) and 2, / Im(z +a^) . , respectively. Then, i f R k ) n ( z ) = [ l ( R n ( z ) ( - l ) k - 1 + ( R ^ t ) + l * ( t ) ) * ) ] (k = l,.2;n = 2,3,. .. ) , (Al . 1 9 ) we take, as we did i n the derivation of ( A l . l l ) , ' s i g n [ l n ( z ) ] [ l n ( z ) / 2 R 2 ^ n ( z ) + i R 2 j n ( z ) l R n(z) < 0, In(z)40,R2^n(z)+0 DR n(z) + i l n ( z ) ] * = f n ( z ) / 2 R 2 , n ( z > + 1 R2,n( z> >' Rn(z)<0,In.(z) = 0, R2^n(z)+0 R l , n ( Z ) + 1 V Z > / 2 R l , n ( Z > > R n(z) > 0, R l j n ( z ) + 0 , S> , R2,n^ z^ o r R l , n ^ z ^ = 0 ! n' 76. Summarizing these r e s u l t s we obtain N(z) ~ % + n(z) (jz §-- . . . (2n-l)/[ (4 n-l)(t / 2 + l/8n + J(t/2)2 + l /64n 2 ) : ] . ) , 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 ( A l . l l ) and (A1.20). The approximations were computed i n [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 re s u l t s . I t l i s t s suitable approxi- mations for d i f f e r e n t subregions of D - . For s i m p l i c i t y the approximations i n (A1.21) and (A1.22) are denoted by S r and C r (r = 1,2,0..) , respectively, where r i s a value of n s u f f i c i e n t l y large to give an accuracy of at least 10 s i g - n i f i c a n t 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 ) . . K B ) : ( , 3 .5 , .. 5 ] V £ °> • 5 ] C10 ( - 4 . 2 5 , - 3 . 7 5 ] -v ; •••/.}. ( 4 . 2 5 , • ; 5 :.' ] - c i o ( - 4 . 7 5 , - 4 . 2 5 ] ', . ( 3 . 2 5 , 5 ] c i o 0.-5, -4.75 ].-•••• .•(. 2 .25 , . 5 . ] c i o ( 2 .25, 5 ]' ( o, 5 ] ; °20 ( - 2 . 2 5 , -1.75'.'] '..•/••. ( 3 . 7 5 , 5 ] c 2 o (-2,75/ • - 2 . 2 5 ] : . ( 2 . 2 5 , 5. ] : c2o (-3-75, ,-2"75 ] , 'v [ . 0, 5 .] ' c2o ( - 4 . 2 5 , -3 .75 ] [ 0, .. 4.25 •] C20 ( - 4 . 7 5 , - 4 . 2 5 ] : •'[•: 0, 3.25 . ] . C20 [ -5 - 4 . 7 5 ] • • [. .0, '/•< 2.25 ] C20. ( - 2 . 7 5 , -2 .25 ] . [ •, 0,' • 2.. 25,' ] C30 ( -75, > 1.75 ] •;•=•• [ 0, ;. .25"-.] ( - 2 5 , ..." -75-] • . l • 0, .. 1.75 ] y - s 5 (-1.25, - , 2 5 ] . ; [ 0 , .75 ] • : S5 (-1.75, -1 .25 ] ; [ 0 , .25 • ] . : s 5 ( . 7 5 , 1.75 ] : ' ( . 2 5 , 3.25 ' ] • : -'• s i o ( - . 2 5 / - 7 5 ] / $ "•( 1-75, 3.25 ] s i o (-1.25,.. - . 2 5 ] ( . 7 5 , 2.75 ] s i o (-1.75,; ' - 1 .25 ]'-•'. ^ ( . 2 5 / 2.25 ] s i o (-2 .25,; -1 .75 ] ."• [ 0, 1.25., ] - : s i o ( 1 . 7 5 , j ' • 2 . 2 5 ] 2.75 . ] ( 1.75, . 2.25 ] •;• < 2 - 7 5 , .'• 5 ] • s i 5 ( . 7 5 , ; 1.75 ] • • ( 3 . 2 5 , 5 : • ] • S l 5 ( - . 2 5 , •'• .75 ] ":; "":'•//. ( 3 . 2 5 , • 5 ' ] • : S15 (-1.25,. - . 2 5 ] : , ( 2.75, S15 (-1.75, - 1 - 2 5 ] " V ( 2 . 2 5 , ' ' ' S15 (-2.25, - i . 7 5 ] ( 1 . 2 5 , 3.75 ] S15 '•/:'; ' 7 8 . . A2 Computer Program for Evaluating; the Saddlepoint 2 Approximation. To evaluate the-..saddlepoint 2 approximation ( 1 . 4 . 2 5 ) , 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 l i s t i n g of a sample run to calculate the approximation ( 1 . 4 . 2 5 ) 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 f i r s t 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 w i l l contain the f o l - lowing information: •'; ./• i ) Cards 1 ,2 and 3 - t i t l e or comments. i i ) Card. 4 - the constant PARA, which may be any parameter that the user 1wishes to vary during ... the problem. If PARA _> 1000 , the program terminates. i i i ) 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 w i l l read 2 . or 1, respectively. (iv) Card 6 - the constant NCUM, which s p e c i f i e s the number of cumulants to be read i n or generated. (v) Cards 6 + 1, 6 + 2,... - (optional) i f Card 5 reads 2, the cummulants w i l l have to be read i n as data, (vi) Card 7 - the number n ; i f n J> 1000 , the program returns to ( i i ) . ( v i i ) Card 8 - the number x ; i f x >_ 1000 } the program returns to ( v i ) ; otherwise, addi t i o n a l values of x are read i n . ... ; . . . 8 0 . . . 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 . n G O TO 1 24 REA0(5,300) (CUMU ),1=1,NCUM) 25 300 FORMAT 13026.16) ; , 2 6 GO TO 222 _ '• . —. ......... ... 27 1 DO 3 1=1,NCUM 28 3 CUM f I ) = CUMUL(I) / 29 77? RFADC5.1071N 8 1 . 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) * P I l , 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) = Q U ) 71 G(2)=U(3)*Q<4)/(STAR**3*6.D0) . . 0 2 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 W R I T E ( 6 , 1 1 0 0 ) ( Z l O C I l t l ' l f 5 ) 85 1100 FORMAT(4X,13HSADDLEPOINT 2/5(1X,D16.9/) ) 86 GO TO 21 0 8 7 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 F K T ) 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) U 4 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) ? l l 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 / 8 6 . ' REFERENCES [ 1 ] Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, Dover Publications, Inc., New York ( 1 9 6 5 ) . [ 2 ] Ahlfors, L.V., Complex Analysis, McGraw-Hill Book Company, Inc., New York ( 1 9 5 3 ) . [ 3 ] Cramer, H., Mathematical Methods of S t a t i s t i c s , Princeton University Press (1945). [4] Cramer, H.', Sur un nouveau theoreme-limite de l a theorie des p r o b a b i l i t e s : Actualites Scientifiques et I n d u s t r i e l - l e s , No. 7 3 6 , Hermann et Cie, Paris ( 1 9 3 8 ) . [ 5 ] Daniels, H.E., Saddlepoint Approximations i n S t a t i s t i c s , Ann. Math. Stat., v o l . 25, pp . 6 3 1 - 6 5 0 ( 1 9 5 4 ) . [ 6 ] De Bruijn, N. G., Asymptotic Methods i n Analysis, North- Holland Publishing Co., Amsterdam ( 1 9 6 1 ) . [ 7 ] F e l l e r , W., An Introduction to P r o b a b i l i t y Theory and i t s Applications, Vol. I I , John Wiley and.Sons Inc., New York ( 1 9 6 6 ) . 7 [ 8 ] Gnedenko, B.V. and Kolmogorov, A.N., Limit Distributions f o r Sums of Independent Random Variables, Addison-Wesley Publishing Co. , Cambridge, Mass. ( 1 9 5 4 ) . [ 9 ] Gurland, J., Inversion formulae f o r the d i s t r i b u t i o n of ratios,- Ann.,.-Math. Stat., vol. 1 9 , pp. 2 2 8 - 2 3 7 ( 1 9 4 8 ) . [ 1 0 ] ' Jeffreys, H. and Jeffreys, B.S., Methods i n Mathematical Physics, Cambridge University Press ( 1 9 5 0 ) . 87. [11] Kendall, M. G. and Stuart, A., The Advanced Theory of S t a t i s t i c s , Vol. I, C. G r i f f i n and Co., London ( 1 9 5 8 ) . ['12] Parzen, E., Modern Pr o b a b i l i t y Theory and i t s Applica- tions, John Wiley and Sons, Inc., New York ( i 9 6 0 ) . •[13] Rubin, H. and Zidek, J. , Approximations to the D i s t r i - bution Function of Sums of Independent Chi Randoirr V a r i - ables, Technical Report No. 1 0 6 , Dept. of S t a t i s t i c s , Stanford University, Stanford, C a l i f o r n i a ( 1 9 6 5 ) . [14] Shenton, L.R., Inequalities f o r the normal i n t e g r a l including a new continued f r a c t i o n , Biometrika, Vol. 4 l , pp. 177-189 (1954). [15] Tiku, M.L. Series expansions f o r the doubly non-central F - d i s t r i b u t i o n , Australian Journal of S t a t i s t i c s , Vol. 7, pp. 78-89, Sydney (1965). [16] Watson, G. N., Theory of Bessel Function, Cambridge Uni- v e r s i t y Press (1948).

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