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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 . S c , Queen's U n i v e r s i t y • o f 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 to the r e q u i r e d  The  t h e s i s as conforming standard.  University of B r i t i s h  Columbia  In p r e s e n t i n g an  this  thesis  in partial  advanced degree a t t h e U n i v e r s i t y  the  Library  shall  make i t f r e e l y  I f u r t h e r agree t h a for  permission  h i s representatives.  of  this  written  M*  rH£Mft-Tt-C$  SePT.'&i*,  Columbia  I agree  that  copying o f t h i s  thesis  by t h e Head o f my D e p a r t m e n t o r  shall  permission.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  Date  gain  Columbia,  f o r r e f e r e n c e and s t u d y .  for extensive  I t i s understood  thesis f o rfinancial  Department o f  of British  available  s c h o l a r l y p u r p o s e s may be g r a n t e d  by  f u l f i l m e n t o f the requirements f o r  that  copying o r p u b l i c a t i o n  n o t be a l l o w e d w i t h o u t my  ii.  Supervisor:  Dr. J. Zidek  ABSTRACT  In t h i s thesis we present two approximations the d i s t r i b u t i o n function of the sum of dom variables.  n  to  independent ran-  They are obtained from generalizations of. asym-  p t o t i c expansions  derived by Rubin and Zidek who  the case of chi random variables. tained from Gurland's  considered  These expansions  are ob-  inversion formula for the d i s t r i b u t i o n  function by using an adaptation of Laplace's method for integrals.  By means of numerical results obtained for a variety  of common d i s t r i b u t i o n s and small values of  n  these approxi-  mations a r c compared to the c l a s s i c a l methods of Edgeworth and Cramer.  F i n a l l y , the method i s used to obtain approxima-  tions to the non-central chi-square d i s t r i b u t i o n and to the doubly non-central  F - d i s t r i b u t i o n f o r various cases defined  in terms of i t s parameters.  iii.  TABLE OF CONTENTS  O  PAGE . 1•  INTRODUCTION CHAPTER I . 1.1  Notation  5  i. 2 '  The Edgeworth Approximation  6  The Cramer Approximation  7  .1.3 1.4 '.• 1. 5  CHAPTER I I . '  5  . NOTATION AND PRELIMINARY RESULTS  • 2.1  •  2.2  CHAPTER I I I .  The S a d d l e p o i n t Method  10  Remarks  20  THE SADDLEPOINT APPROXIMATIONS  21  Asymptotic Expansions  '  • 33  ..The. L a t t i c e Case  37  COMPUTATIONS  3.1'  Remarks on the T a b l e s  3-2  " C h i Random V a r i a b l e s ; "  21  "  37 39  3.3  The E x p o n e n t i a l 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 P r o b a b i l i t y Law  . 49  . . . .  3.6  The Uniform P r o b a b i l i t y Law  3.7  Remarks  54 . 5 8  CHAPTER IV.  OTHER APPLICATIONS  59  '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 - D i s t r i b u t i o n  63  4.3  Remarks.  70  • 4.1  :  APPENDIX Al  .  '  ' 71  1  • Computing  A2  N(z)  Computer Program f o r / E v a l u a t i n g  >  71  the  78  Saddlepoint 2 Approximation  REFERENCES  .  ••  86  ACKNOWLEDGMENTS  * The author wishes t o thank Dr. J . Zidek f o r suggest i n g the t o p i c of t h i s t h e s i s and f o r generous given d u r i n g  assistance  its.writing.  The f i n a n c i a l support o f the N a t i o n a l Research C o u n c i l o f Canada and of the U n i v e r s i t y of B r i t i s h Columbia is also gratefully  acknowledged.  INTRODUCTION  . I t i s o f t e n n e c e s s a r y to approximate the  distribution  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 c o n v e n i e n t l y be c a l c u l a t e d .  cannot  For example, i n the e v a l u a t i o n of  e r r o r p r o b a b i l i t i e s i n some t y p e s . o f communications systems or sometimes i n the d e t e r m i n a t i o n of the power f u n c t i o n of a 1 P(—  l i k e l i h o o d r a t i o t e s t , the p r o b a b i l i t y  n  where the  X^  ( i = .1,2,....)  are independent  •is r e q u i r e d f o r some v a l u e s of  515). was  random v a r i a b l e s , 1  ([3L  to Edgeworth  available for this pp.  228-229-  Another i s t h a t of Cramer ( [ 7 ] , 'p. 5 2 0 ;  x  i s p e r m i t t e d t o depend on  [4]),  p.  which  even i n the  n  When the c h a r a c t e r i s t i c f u n c t i o n tic  type  [7],  designed t o p r o v i d e an accurate approximation  case where  ,  1  x  A well-known approximation of problem i s due  n  £ X. < x) i=l  of the  statis-  i s known, i n v e r t i n g the F o u r i e r transform e x p l i c i t l y i s  often impossible, while a numerical i n t e g r a t i o n routine i s z  too time-consuming f o r procedures e s p e c i a l l y when  U(t)|  consider two  gested by Rubin and Zidek [13], :  of  approximations  n  accuracy,  i s not r a p i d l y d e c r e a s i n g as  In t h i s t h e s i s we  o f the sum  r e q u i r i n g high  independent  |t| - » .  approximations,  sug-  t o the d i s t r i b u t i o n f u n c t i o n random variables'.  i s called, in [13],  a saddlepoiht  Each of these approximation  i n keeping w i t h the terminology used by D a n i e l s [5], seems t o have been the f i r s t t o i n t r o d u c e i n t o the  who literature  of p r o b a b i l i t y approximation theory the method upon which the approximations i n [ 1 3 ]  are based.  Whereas i n t h i s t h e s i s  we  are concerned w i t h problems r e l a t i n g t o d i s t r i b u t i o n f u n c t i o n s , the work of D a n i e l s p e r t a i n s t o d e n s i t y f u n c t i o n s . The e f f o r t s of Rubin and Zidek [13] are d i r e c t e d toward  the problem  b u t i o n f u n c t i o n of are independent  of f i n d i n g an approximation to the ( | Z-j^ | + . . . + | Z | )  random v a r i a b l e s and each  i s normally d i s t r i b u t e d w i t h mean  0  Z.  distri-  , where the  {Z } i  ( i = l,...,n)  and v a r i a n c e  1  Each of t h e i r approximations i s r o u g h l y e q u i v a l e n t , i n terms of r e q u i r e d  computer time, t o those of Edgeworth and Cramer.  In [13] i t i s shown f o r the problem  considered t h e r e , on the  b a s i s of n u m e r i c a l r e s u l t s , t h a t one o f . t h e approximations i s s u p e r i o r to e i t h e r of the two the case  n = 10  c l a s s i c a l methods.  Even f o r  , where f o r values of the arguments con-  s i d e r e d , the o l d e r methods y i e l d an accuracy of at most s i g n i f i c a n t f i g u r e s , i t g i v e s r e s u l t s accurate t o f i v e ficant  two signi-  figures. In Chapter I of t h i s t h e s i s a l l of the  mentioned  above are presented.  approximations  .Also given i s an i n v e r s i o n  mula, d e r i v e d from t h a t of Gurland [9],  for-  which forms the b a s i s  f o r the s a d d l e p o i n t approach. The  contents of Chapter I I c o n s i s t of p r o o f s that the  f o r m a l expansions d e r i v e d i n [13] are i n f a c t asymptotic ex- , pansions.  In [13] i t i s suggested t h a t these r e s u l t s might  be  obtained by u s i n g an a d a p t a t i o n of the argument given by D a n i e l s [5], which i s based on the method of s t e e p e s t descent. Here we.give f o r the case of n o n - l a t t i c e  random-variables  3.  simple and  d i r e c t p r o o f s which use Laplace's method f o r i n t e g r a l s  s p e c i a l f e a t u r e s of the present problem.  i n one case, an expansion  i n powers o f  The r e s u l t s a r e ,  n *- , and,  i n the other, as i n D a n i e l s ' case f o r d e n s i t i e s , an  expansion  i n powers o f n""*" The with those  approximations  of Edgeworth and Cramer f o r a v a r i e t y of s p e c i a l  cases on the b a s i s of numerical of  n  found  .  computations f o r s m a l l values  The r e s u l t s are q u a l i t a t i v e l y the same as those  i n [13]  presented  compared, i n Chapter J>,  are then  f o r the s p e c i a l case  considered t h e r e , which i s  f o r completeness i n s e c t i o n ( 3 . 4 ) . The r e s u l t s given i n Chapter 3 are obtained w i t h the  i n t e n t i o n o f comparing the v a r i o u s approximations earlier. 3.7,  Except  f o r the cases  considered  d e s c r i b e d j- •  i n s e c t i o n s 3.4 and  the d e s i r e d v a l u e s can be found w i t h reasonable  from e x i s t i n g t a b l e s .  In Chapter 4 ,  accuracy  some a d d i t i o n a l a p p l i c a -  t i o n s - o f t h e " s a d d l e p o i n t method"'are considered. • These i n v o l v e the n o n - c e n t r a l chi--square. d i s t r i b u t i o n f o r l a r g e values of i t s n o n - c e n t r a l i t y parameter,; and the d o u b l y n o n - c e n t r a l :  F - d i s t r i b u t i o n f o r s e l e c t e d s p e c i a l cases d e f i n e d i n terms of i t s f o u r parameters.  In the l a t t e r case  e x i s t i n g t a b l e s are inadequate,  particularly,  and the s a d d l e p o i n t  approxi-  mation may be o f p r a c t i c a l value. Two appendices are s u p p l i e d .  In the f i r s t  i s given  a method of e v a l u a t i n g the normal d i s t r i b u t i o n f u n c t i o n f o r e i t h e r 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 [ 1 3 ] .  In. Appendix  2,  we d e s c r i b e a program w r i t t e n i n FORTRAN which i s . s u i t a b l e  f o r n u m e r i c a l l y e v a l u a t i n g the better, of the two approximations.  saddlepoint  5. CHAPTER I  NOTATION AND PRELIMINARY RESULTS  Notation.  1.1  Let  {X-j^Xg,...}  be a set  of independent,  identically  d i s t r i b u t e d random v a r i a b l e s , each h a v i n g d i s t r i b u t i o n f u n c t i o n 2  F and  , density function  f  , mean  characteristic function  co(t) = E [ e  l t X  = J "  where  cp  u = 0 ; that  , variance  (-»<t<«0 i t x  dF(x)  ,  i = /^T A  ¥ e assume the moment g e n e r a t i n g f u n c t i o n of e x i s t s on a non-degenerate M  ...  ,  is,  ]  e  a  Let  K be the  M(it)  interval  and denote  cumulant g e n e r a t i n g f u n c t i o n .  = cp(t)  and  ,  / •  K(t)  (a,b)  = l o g M(t)  -~<t<»  X^ i t by  Then  ,  • .  We take the domains of  M and  complex plane given by  [z  K  to be the subset of  the  : a <.Re(z) < b}  The d i s t r i b u t i o n and d e n s i t y f u n c t i o n s of the dard normal d i s t r i b u t i o n w i l l be denoted by  N  and  n  stan,  res-  pectively.  of  . Let  n  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  E  (n = 1 , 2 , . . . )  ,  X . / ( J ? T G )  i=l  that i s ,  ;  1  where . P  denotes the , n r - f o l d . c o n v o l u t i o n • of  P. .  More  generally, i f : u ^ 0 , F and P. w i l l denote the d i s t r i b u n tion functions of E (X. - u ) / (,/n a) and (X. - ^) , n  i=l r e s p e c t i v e l y , while t i o n of  1.2  M  1  w i l l denote the moment g e n e r a t i n g func-  ( X - u) . i  The Edgeworth  Approximation.  THEOREM 1 . 2 . 1 . r  th  absolute moment of F  F  1  I f l i m sup |co(s)| < 1  , and the  \s\-**  exists,  f x ) = N(x) + n(x) Y n"* • k=3  then  k + 1  R. (x) + o ( n ' *  r  +  1  ) (1.2.1)  (n-»)  uniformly i n x depending not on  n  on  u-^,. . .u  , the f i r s t  , _or otherwise on  PROOF. The  See F e l l e r  P  r  R-^,. . . ,R  r  , ach _e  moments o f F  , but  _or r  [ 7 ] , p. 515.  s e r i e s i n ( 1 . 2 . 1 ) i s known as the Edgeworth  expansion o f F n R (k = l , . . . , r ) k  , f o r some polynomials  .  The c o n s t r u c t i o n o f the p o l y n o m i a l s ,  , 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 e x p l i c i t l y , this expansion f o r  F  n  is  F (w n  n  = N(w ) n  - rT*rif N( (w )] 5)  n  +  n -  1  [ ^ N ^ ) ( w  )  n  ^ x | N< >(w )] 6  +  n  3 r  ?  5I x  ^n  iV  fi  ^  2100 ,2,  + "TOT 3 4 X  77 '"3*4  ; T  X  N l  ^ n w  ~"9J"'\5  ;  2  „(10), ( n) ;  w  N(  8 )  (W  v , 154 00 \4 ..(12 -121- *3 (wj]  )  N  +  N  -  }  (1.2.2),  where n  t h  of  w  n  = (x -  nu)/(Vn  cumulant of F  ),  , \ = ct /a  a)  (a  n  n  n  and  n  denoting the  denotes the i  t  h  derivative  N . In practice, i t i s not advisable to go beyond the  second or t h i r d term of the series (1.2.2), as a well-known disadvantage of the Edgeworth expansion i s that i t then tends to give negative values f o r values exceeding  1.3  1  when  x  F ( ) when x  n  x  i s small or  x  depends on  i s large.  The Cramer Approximation Cases occur where, i n ? ( ) > x  n  n  ,  as "when computing  P[ X _< y] = P («/n y/a) n  Then (1.2.1) w i l l f a i l t o h o l d . F (x)  and  n  N(x)  converge t o  Since 1  p r i a t e c r i t e r i o n of the accuracy relative error.  as  w = 0)  (with  .  i n t h i s case both • n -» »  , a more appro-  of the approximation i s the  In p a r t i c u l a r , we would l i k e . t h e r e l a t i o n  1 - F (x)  (1.3.1)  n  1 - N(x).  to h o l d when both  x  and  "  n  tend t o i n f i n i t y .  This  relation  i s not t r u e g e n e r a l l y , s i n c e , f o r example, i n the case of the symmetric b i n o m i a l x > Jn  .  distribution,  1 - F (x) = 0 n  for a l l  The l i m i t i n (1.3.1) does h o l d i f x = o ( n  1//  t h i s being a consequence o f the f o l l o w i n g more g e n e r a l  ^)  ,  result.  I t i s due t o H. Cramer [4] and was g e n e r a l i z e d t o v a r i a b l e components by F e l l e r  ( [ 7 ] , p. 524).  L e t the f u n c t i o n  •  \  z  where-  3  h  V  be d e f i n e d by the equation  X(z) = K(h) - h K ^ ^ h ) + -|z  i s obtained  as a power s e r i e s i n  (1.3.2),  2  z  by i n v e r t i n g the  series  oz =  V  r=2  =  Equation  (1.3.2) i m p l i e s  :  ( ri). " hr  r  (h)  1  (1-3.3)  X ( z ) =• X / 6 + ( X / 2 4 - \|/8)z 4  5  + ( x ^ / 1 2 0 - \j\ /12 h  THEOREM. 1 . 3 . 1  (Cramer). 00  number  h  such t h a t  Q  j  )  0  on  n  .  Let x  , such t h a t  +...••..  2  (1.3.4)  Suppose t h e r e e x i s t s a  _  e  dF(x) e x i s t s f o r a l l  CO  —  h e (-h ,h  + \j/8)z  _be a r e a l number, which may. depend  x > 1  and  x = o(n ~)  as  2  n - «  Then  v  ,  n  m^y'*  For  HJL-mi  1 - F ( x ) « [1 - N(x)] exp [£-  ^ ) ]  , the c o r r e s p o n d i n g r e l a t i o n i s  F ( x ) » N ( x ) exp [j^X n  ( ^ ) ]  (n -  [  l  +  0  ( ^ ) ]  p. 517.  When the Cramer approximation i s a p p l i e d , t i e s may be encountered i n the i n v e r s i o n  sible to invert  (1.3-3)  (1.3.3)*  t o converge.  occurs are given i n s e c t i o n  p r o x i m a t i o n can s t i l l be a p p l i e d  difficul-  of the s e r i e s  the s e r i e s given i n ( 1 . 3 . 4 ) may f a i l  Examples where.this  , (1.3.5b)  '  See F e l l e r [ 7 ] ,  PROOF.  because  (  (1.3.5a)  (n - .)  x < -1  0  +  3-3.  The ap-  i n these cases i f i t i s pos-  algebraically.  10.  1.4  The Saddlepoint Method.  We now present two s a d d l e p o i n t approximations t o F  n  obtained by Rubin and Zidek [ 1 3 ] .  They are based  on an  i n v e r s i o n formula d e r i v e d i n Lemma 1 . 4 . 1 from the Gurland [9]  i n v e r s i o n formula.  1  - F (  I t a s s e r t s that -e T .. lim { f + f }  ) = * + lim  -ixt M (it)dt n  (1.4.1)  LEMMA 1 . 4 . 1 . function  M  Assume t h a t the moment g e n e r a t i n g  e x i s t s i n the r e g i o n  (z : a < Re(z) < b] .  Then  s io:u u^ . -e  .T  p  +  T  e  = W - I_l  Y  e-<  c + i u  >  -ixt  M ( i t ) d t  MV+lu)!^  x  -  t  sign ( = ) • '  (1.4.2)"  f o r every r e a l number  sign(t)  c ^ 0  - 1 0 1  V PROOF. • Suppose  such t h a t  a < c < b  3  where  t < 0 t = t > b > c > 0  I 0 0  , and consider the l i n e  i n t e g r a l i n the. complex plane of the. f u n c t i o n  11. f(z) = e"  M (z)/(27riz)  Z X  n  a l o n g the contour  I = I  constants  € (e < T)  where, f o r f i x e d p o s i t i v e  U J  2  =  , e < Im(z) <_ T]  ,  : Irn(z) = T  , 0' < Re(z) _< c}  ,  {z :  Re(z) ---- c  =  h  =  5  =  U  6  =  (z : z =  z  :  Im(z) =  :  Re(z) = 0  €6  -T  , -T < Im(z)  < T}  , 0 <_ Re(z)  < c}  f J  I  f(z)dz = 0  ie  2  ^  2  J  c f | exp o  '•  |E(exp[(y+iT)(X  < E(exp[y(X = ^ ( y )  ,  J 2  (-x(y+iT))  M (y+iT) | dy • |y+iT n  1  1  ' + . . . +  + ... +  X )])| n  X )]) n  •  Hence, If  f ( ) d z | <.1 2  t ^ - | ( l  - e" )]  2 A =  +...+  e]  '/  [ ^ ( y + i T ) |; =  where  2  .  . |  J  •/  -  I  Now,  • i t - f ( z ) dz I  since  <_ -  , -T < Im(z)  i  Then  and  Re(z) = 0  :  h  T  T  1  max M (y) o<y<c n  , and t h e r e f o r e  c x  ,  •  Ig  12. lim  f  lim j i  Similarly,  T—  f ( z ) dz = 0  f(z)dz  = 0  4  J  Consider  f(z)dz x  .  By the r e s i d u e theorem,  6  , v f(z)dz = 1 i :(e) v  where e  c(e)  i s a c i r c l e w i t h centre at the o r i g i n ,  radius'  , 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 TT/2  lim  :e-o\  .  f J  f(ee  -77/2  ..  fl  )eie  i e  that  e-o  -TT/2  .  a  dfi = l i m f '  1 B  f(ee  )eie  i e  dfl  i p  TT/2  (1.^.3)  Prom t h i s i t f o l l o w s that the  desired  lim  f(z)dz  = \  , and  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 (e n  cos  fi+  e i s i n e)d  -TT/2  1 -75= W  =  f J  '  e  -xefcosP + i v  s i n ;9 ) w n /  „  ,  .  .  (e cose + e 1 s i n e ) d e |  M  TT/2  1 |W  TT/2  f J  -TT/2  [- e  x e  (  c o s  fi  +  1  s i n  *)  M  n  (e  cose +-  e  i sine)  13. _ -x€(-cose + i s i n e ) n _ e  M  < 1_ J  | -  V 2  x e c  o  s  e  9  (  £  c  o  s  0  +  g  ±  s  i  n  s  )  ]  d  0  |  M (e cose + e i s i n ) - e n  X e  C  O  s  e  9  -TT/2  x M (-e cose + e i . s i n p ) | d e n  By;the Legesgue bounded tity  converges t o  established. f  f(z)dz  For  0  convergence theorem, the l a s t  as . e -• 0  and the r e s u l t , (1.4.3), i s  Combining the r e s u l t s f o r J  , and  a < c < 0  f  f(z)dz  quan-  f(z)dz  ,  , we o b t a i n (1?4.2) f o r c > 0  .  , the p r o o f i s s i m i l a r .  We s h a l l make use of the i n v e r s i o n formula  (1.4.2)  i n the form,  1 1  -  F  N  ( ^  x/a)  -  -1(1 - s i g n ( c ) ) +  l'  lC=  exp[-nx(c+iu)+n logM(c+iu)]  TJ^-.J — 00  where we s h a l l choose  c  i n the i n t e g r a l ; t h a t i s ,  t o be a s a d d l e p o i n t of the exponent c  i s the s o l u t i o n of the equation  d [-nxz + n l o g M("z') ] = 0 dz  (1.4.5),  or  t^ When  c = 0  1  - *  t ^ - v -  , the i n t e g r a l i n (1.4.4) w i l l be understood t o  mean that of the Gurland i n v e r s i o n formula. D a n i e l s [5] showed t h a t root under f a i r l y g e n e r a l  (1.4.6) has a s i n g l e r e a l  conditions.  14. THEOREM 1.4.1  (Daniels).  M (t) = e ^  converges f o r r e a l 0 < Cg •<_ <° •  f  =  K  t  e  Assume t h a t  dP(x)  t x  _in - c ^ < t <  , where  0 < c-^ _< <» ,  Suppose  F ( x ) = 0' , •  x < a  0 < F(x) < 1 F(x) = 1  where, p o s s i b l y , (i)  ,  a  a < x < b  ,  b < x  a = -» and  ,  o_r  b = »  exists for a l l real  ,  _or both.  b o are f i n i t e  ,  Then  i f and only i f K ( t ) t  , and (1.4.6) has no  r e a l r o o t whenever  x i (ii)  f o r every  [a,b]  ,  x € (a,b)  , where  - « < a < b < » ,  there e x i s t s a unique simple r o o t (1.4.6)., from (iii)  and  K^?~\t)  x. = a  f o r every  to  of_  continuously  ,  x e ( a , b ) , where  be i n f i n i t e , c  x = b  increases  c  ?a  there e x i s t s a  and  b  may  corresponding  in  lim ' t—c^  (-c-^Cg). i f l i m K ^ ^ t ) = b and (1) t-»+c K ( t ) = a (these c o n d i t i o n s are s a t i s 2  v  f i e d a u t o m a t i c a l l y unless finite).  '  a  or  b  _is i n -  PROOF. Since c e (-c-^Cg)  See Daniels '[5]. K(t) converges f o r  -c-^ < Re(t) < c  , and  2  , K(t) has a power series expansion about  t =c  with a non-zero radius of convergence, and hence a. uniformly convergent series expansion f o r a l l t for  some  p > 0  .  such that  Then, for values of n  11 - c | _< p  i n some neighbour-  hood of the o r i g i n we can write 2 -nx(c+iu) + n K(c+iu) = -nxc + K(c)n - n u ( 2 ) ( ) "5 K  +  c  £ K^ (c) ( i u )  n  r)  r=3  (1.4.7)  r  rJ  Let  a * - [K< > ( C ) ] *  , '  2  b  r  = K< > (c) i / ( r J a * )  a  r  = (-i) /(ca*)  r  r  r  r  K(c,n,x) = J2T I(x,n) =  ^  e"  J "  E  (1.4.8)  ,  r  -  N  n x c  X  (  +  C  n K  +  I  < >  ,  c  U  )  +  N  K  (  C  +  I  U  du c+iu  )  Then, proceeding formally, c  m  I(x,n) = K( c,n,x) ./2TT  J -»  e"  rlh  n  v  K  ;  r.'  (  c  )  (  i  u  )  ca  j  (l  -na u 2~  i y ,-1 n v b (y/,/rI) ;^) e r=3 r  ^ K l c ^ x )  e  + C f f  r  du c+iu  _2 y  /2  dy  ca ./27m  (1.4.9)  16. Now,  and  /  __iy__ -i  \1 +  C a  */jj ) •  eo  n i b e r=3  2  »  v  =  +'  1  (y/./n)r v  I  a r  • (y//n)  = 1 + n + n"  _^  • (b^y + # b^y ) 6  b^ y^ + n  2  (b y  V 2  IyI < c a V n , • (1.4.10)  1  4  + b b y  5  5  5  7  4  + £  b^y ) 9  + n- (.b y +[i-bf b3b ]y -^b b y 4ifb y 6  2  8  +  6  4  4  + ...  Hence,  10  2  5  (1.4.11).  _^ r=3  */n  m=0  ,/n (1.4.12),  where ;  d  0  ( y )  =  •  1  2  d (y) = a y 2  2  d^(y) +  ±  3  4 2 6 + ( b + a-jb^y + | b^ y + ( b - K a . ^ + agb^ + a ^ b ^ y  4  4  2  + (^b  8  2  M SM>.\;  + ^ b ^ y  2  . v : } K . . •;•'• ,  :  and, i n general, while:•d (y)  d  2k-l^ ^  As odd powers of .y form of •.'d _ 2k  y  i s  3 X 1  o  d  d  1  (y)  1  0  y  + (|b  +  2  b^y  +  1 2  y  ,  (k = 1 , 2 , . . . )  vanish upon integrating, the e x p l i c i t (k = 1 , 2 , —  )  b ^  (1.4.13),  ; '•''•;  polynomial i n  an even polynomial i n  2k  6  6  + ta b )y  k  ! '  4  = a y  abb  '  i s not required below.  It w i l l be shown i n the following chapter that  12  • 17. I(x,n) ~ ( j;*"'*)  ,  K  i s an asymptotic expansion.  d  2m  f  =  n  2m  V^=;  d  T  (1.4.14)  Here,  ( y ) 2m^)  ^  d  (1.4,15)  — CO  1 ( r e c a l l that  _  n(y) =  2  / 2  e ^  )  , and the s e r i e s i n (1.4.14)  i s obtained f o r m a l l y 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 i n the e x p r e s s i o n obtained from (1.4.9) by r e p l a c i n g the f i r s t two f a c t o r s i n i t s i n t e g r a n d by t h e i r s e r i e s The f i r s t  d d  = a  2  4  few c o e f f i c i e n t s i n (1.4.14,) are  + 3(b  2  expansion.  + a-jb^) +  4  b|  (1.4.16)  3a^ + 15(bg + a-j^b^ + a b ^ + a^b^) + 105(tb  =  2  + a b b ' + |a b|) +'945(t'b b| + ^ b ^ ) 1  3  4  2  (r ;=• 1,2,...)  d  o  d , 4 d  +  2  =  +  4  E x p l i c i t l y , equations (1.4.16)  are, w i t h  K^ ^ r  + b^b^ b^  .  (c) = K°  ,  1  :  = -(ca*)"  2  = 5(ea*)-  4  35 "2T  i|  *-8,l2 ^T6 4  + a*- '^ 4  + | a*" (- ^ 6  1  K  a  K  -  +  5  Kg + ^  1  „  T o 3 5 " "2^ 3 4 K  K  -IT  K ) -  K  K  K5- ^ c  1 +  4(o*r  ^2 3 K  T6  (1.4.17)  +  ^ c  *-10  2v , 3 5 }  6  a  18. , 1 2 x (- ^ 1 ^ +  3s  1  ^K  5  ) +  385  3 ^ a  *-12_4  .  Thus, . x . F (>/fx_). ~ * ( l - 8 l g n ( c ) ) n  +  e  -™  c  +  n  K  < >(l c  ^  +  d ^ )  +  072701X2  n  (1.4.18) , where  c  i s obtained from  (1.4.6) and  d  and  g  d^  from  (1.4.17). Let  us now r e t u r n t o (1.4.4) and by an a l t e r n a t e  argument a r r i v e a t another On l e t t i n g  p = c JnK^  s a d d l e p o i n t approximation  and  b ' = b /i  t o F „ .. n  (r = 1 , 2 , . . . )  r  r  , we  have, by r e g r o u p i n g the f a c t o r s i n the i n t e g r a n d of (1.4.4), CO r  "nrbVr/  1  P  =  • * ( l - s i g n ( c ) ) + K(;c,n,x)  f '. n ( u |  / rrr?  e r=3  (1.4.19) CO  ,v  S b ,{iu) /;/H " • r=3 - v: ".  But  e  r  r  2  ;  S (y) r  - » •-, £ g ,(iu)(/-J  = ( '- 0,1,2,...)  are d e f i n e d i n the obvious manner  r  equation (1.4.11). • ' . Define Q (p) k  Q(p)=J  , where  T  from  • '/ (k = 0,1,2,...)  by  n{u)_ ( i u ) du  (1.4.20).  k  -» p+iu  Then, V  p  >  =  nTpJ  fi ^ )!  +  1  Q (p) = 1 - p Q (p) x  0  ;  0  ' (P)) N  (1.^.21), (1.4.22),  19. . Q ( p ) (k = 2 , 3 , . . . )  and  s a t i s f y the r e c u r r e n c e  k  Q^p) Q  2  k  = "P ^ 2 k - l (  _ (p) 1  (  p )  k  =  1  (1.4.23),  > »'") 2  = (-i) " (2k-3)(2k-5)...(3)(l)-pQ k  formulae  1  2 k  _ (p)  .  2  (1.4.24). 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 integration,  a result  of the form k  oo 1  " nV^r) P  ^(l-sign(c)) + K(c,n,x)  =  I  h  k  Vn  k=0  n  (1.4.25),  ( p ) ( — i  where h (p)  =  h-^p)  = ^ a*  h (P)  =^74 a * ' K Q ( p )  h (p)  =  Q  0. (p)  5  1  K Q (p) 3  a* K  ,  3  4  _ 5  4  a*" K Q (p)  +  K Q (p) 5  "5  -*_Q  + i296 °  +  _ 3  4  2  - M  ,  0  6  + ^  5  2  (1.4.26)  ,  6  a*- K K Q (p) 7  4  3  7  , .  3^( ) p  '  Y5o * " 6 % ( ) (TT55 4 T!O 3 5 *" 8 1 * 1 0 2 _ / \ , 1 *-12 _4_ , v THZ8 0 3 4 loi + 3110^ K Q (p) , p )  =  a  6 K  v  K  I  K  p  A  Q  K  +  p )  +  0  k  k  using equations  (k = 0 , 1 , . . . , 1 3 )  (1.4.21),  (1.4.22),  are r e a d i l y  ( 1 . 4 . 2 3 ) and  ) o  8 q  I  3  and where the , Q ( p )  K  1  2  obtained  (1.4.24).  The a p p r o x i m a t i o n o b t a i n e d from ( 1 . 4 . 2 5 ) by d e l e t i n g a l l terms involving  h ^ (k _> 5 )  2 approximation.  w i l l be r e f e r r e d  t o as the  saddlepoint !  ( p  20. 1.5  Remarks  1  T h i s l a s t approximation w i l l be shown, by n u m e r i c a l means i n Chapter  3* t o be s u p e r i o r f o r s m a l l samples t o both  the Edgeworth ( 1 . 2 . 1 )  and the Cramer ( 1 . 3 . 5 )  approximation,  as w e l l as t o the s a d d l e p o i n t 1 approximation c = 0  f a i l s for  (1.4.18),  .  A f a m i l y of f o r m a l s e r i e s expansions can be obtained i n the manner of ( 1 . 4 . 2 5 ) any v a l u e i n  (-c-^Cg)  .  Hence, i f the s o l u t i o n  c  c = 0  proved t o be  of ( 1 . 4 . 6 )  is  0  s a d d l e p o i n t 2 s e r i e s must be i d e n t i c a l .  moves away from  0  mation d e t e r i o r a t e s . f a c t that  c  of . 1 - P  by l e t t i n g  In p a r t i c u l a r ,  Edgeworth s e r i e s , which Cramer [ 3 ]  and  The  the  asymptotic.  , the Edgeworth However, as  explanation f o r this l i e s  As D a n i e l s [ 5 ]  ':. z =. c+iu,,  c  i n the  as i s well-known,  through  (see, f o r  has argued, the path of i n t e -  < u .<<»}  will  closely  approximate  the path of ,'steepest, descent l o c a l l y i n ' the' r e g i o n near s a d d l e p o i n t , from which the 'Only' asymptotic the i n t e g r a l comes.  take  yields  any s a d d l e p o i n t there i s a path of s t e e p e s t descent  g r a t i o n . [z  c  , the q u a l i t y of the Edgeworth a p p r o x i - •• •  i s a s a d d l e p o i n t , and,  example, [ 6 ] ) :  which  the  contribution.,to  21.  CHAPTER I I  THE SADDLEPOINT APPROXIMATIONS  In t h i s chapter we prove that the expansions  given  I in (1.4.18)  and ^ 1 . 4 . 2 5 )  are asymptotic.  To do so we use  L a p l a c e s method f o r i n t e g r a l s and thereby achieve a more 1  s t r a i g h t - f o r w a r d p r o o f than that p r o v i d e d by the method o f .steepest descent  (see, f o r example, Watson [.16]).  Further-  more,, t h i s method i s r e a d i l y adaptable t o other a p p l i c a t i o n s • i n v o l v i n g : p a r a m e t e r s ' d i f f e r e n t from, n  which tend t o 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 ) , , f o r  I(x,n)=^f ° e  -»  Let  iu -1 g(u) = ( 1 + — ) "  interval power  (-R,R)  e  -  n x  .  (  c + i u  >  Then  for a l l R  +  7  n  c^ 0  n K  (  c + i u  ,  )_du_ c+iu  (2.1.1).  g(u) i s a i n t e g r a b l e over the  and i s equal t o the convergent  series  g(u) = 1 - AH + (IH) _ (IH) 2  i n the i n t e r v a l  [-6^,6  ]  , where  6^  3 +  <c  ... .  {2  .1.2)  22. As we s h a l l show subsequently, the a s y m p t o t i c a l l y dominant c o n t r i b u t i o n t o the i n t e g r a l i n (2.1.1) comes from that p o r t i o n of i t which i s over that  g(u)  ,n~^~^].  [-n  In order  be adequately r e p r e s e n t e d over t h i s r e g i o n by th.4 1/3  f i r s t few terms of i t s s e r i e s expansion, in  comparison  to  c~^"  .  n  must be l a r g e  Since i n the case of the saddle-  p o i n t - 1 approximation we assume  g(u)  i s so r e p r e s e n t e d ,  i t tends t o g i v e u n s a t i s f a c t o r y r e s u l t s when and  c  i s near  0  ,n . i s moderate.,-,; The, s a d d l e p o i n t 2 approximation l e a v e s 1  g . i n i t s o r i g i n a l form and tends t o give good r e s u l t s even when / c ' i s ' s m a l l . ' However,, i t i s somewhat more complicated than the s a d d l e p o i n t 1 approximation. ' : Let CO  I a ur r=2  h(u) =  ,  |u| < 6  ,  r  (2.1.3) h (u)  = - i x u + K(c+iu) - K(c)  1  where  a  = K^ ^(c)  i /rJ  r  r  ( r = 2,3,...)  r  p o s i t i v e constant.  ,  and  6  i s some  . /  D a n i e l s [ 5 ] showed t h a t i f the moment g e n e r a t i n g ' : - c ^ < Re(z) < c ]  M  where  i s the l a r g e s t such r e g i o n , one r e a l r o o t  D  exists i n a region  D = [z  function  K^^(-z) - x = 0 As  M(z)  e x i s t s , and i t s a t i s f i e s  i s analytic i n  D  2  -c-^ < c < c  of 2  ,  M(z) = M(c) + M ^ ( c ) ( z - c ) + M ^ ( c ) ( z - c ) l )  c  2 )  21  2  + ...  (2.1.4)  23. converges i n a c i r c l e w i t h non-zero r a d i u s of convergence Therefore,  p < R  some p o s i t i v e  .  s e r i e s expansion about  [z  {z  (2.1.4) converges u n i f o r m l y on  ; | z-c | _< 6 1  Hence, z = c  f o r some  2  R  : |z-c| _< p} f o r  -log M(z) = K(z) has a power which converges u n i f o r m l y on|  &  2  > 0  .  Thus  h- (u) L  has a  power s e r i e s expansion which i s u n i f o r m l y convergent f o r i u j >< & . \ that i s , y  2  h ( u ) = h(u) x  = al u  2 2  +  some r e a l  r  ( |u| < 6  )  6 = min(6- ,6 ) L  2  . If  ,  (2.1.6(a))  icp(t)i ' dt < j  j > 1  (2.1.5).  v  '  Let  00  for  a u ,.  r=3  LEMMA 2, 1.1.  I  £  ,• then  -6rnh,(u) j g(u) e du =.0(n- ). , l V  M  (2.1.6)  and [  for  6  g(u) e  each p o s i t i v e i n t e g e r  PROOF.  I f the  nh, (u) 1  du = 0 ( n " ) M  M  [X ]  have a. l a t t i c e d i s t r i b u t i o n ,  ±  ,. ( i = 1,2,. . . ) M(c-fiu) M(c)  <  p <  1  , do not  24. since  |u| > 6 > 0  (Daniels [ 5 ] ) .  C o n d i t i o n (2.1.6(a)) i m p l i e s , .  03  J  a f t e r a change of v a r i a b l e , t h a t some  j _> 1  , say  k  | M ( c + i t ) | dt < » f o r -oo ^ oo nh-.(u') L = ""TM J g( ) n J  . • Writing  -  u  e  d  u  N  we o b t a i n  !L ! N  < -TM . f  n  for  some constant  P  1. .  <  "  k :  . | M ( c  A > 0 r" J  Similarly,  .  n  M  n  i( )  n  , since  u  du = 0(n  M  )  There e x i s t s a p o s i t i v e for a l l  |u| _< 6^  constant  where  p  6^ i s  constant.  PROOF.  =  du - A . n / p -  k  l i m |L I = 0 n^»  g(u) e  such that. Re{h(u)} _< -pu  .h(u)  Thus  5  LEMMA 2.1.2.  some, p o s i t i v e  i u ) |  +  E x p l i c i t l y , when  K ^ ( c ) (2r)J  S r=l  (-lj  r  u  Therefore,  |u| < 6  ,  h  i s g i v e n by  + i E K< \c) r=l" (2r+l).'  2 r  > l )  2 r + 1  r  u  2  r  +  /  Re{h(u)} = -K  2  + K^ u  2  - Kg u" + . ..  u where is  K =K^ ^(c) 1  i  continuous,  (i'=2,3,...)  and equals  minology and r e s u l t c o n t i n u o u s l y from  -Kg  at  .  Now,  u = 0  ( i i ) of Theorem 1.4.1, x = a  to  x = b  .  .  Re{h(u)}/u  Using the t e r -  K^^(t)  Hence,  2  K  v  increases ;  ( c )> 0  .  1  25.  Select  €  s u b j e c t t o the requirement  Kg > e > 0  By the d e f i n i t i o n o f c o n t i n u i t y , there e x i s t s |u| _< 63  that i f  ^  2  Thus,.; • Re(h(u)} _< -pu  p  + e < 0  for all  |u| _< 6^  and f o r some  ' . (Prom now on, l e t ' 6 = min( 6^, 6g, 6^)  v ... \ F i r s t expand.  g(u) exp(n  S a u ) = g(u) exp^nu^ r  •  fora l l  P(nu ,u) = 5  |u| < 6  .  1  nu  and  y  u  ,  (2.1.7)  J  J  ( i = 0,1,... ; j = 0,1,...)  are independent  n. and u ' . In order t o approximate  sums, we r e s t r i c t |nu | _< 1 5  tervals: 6  6 = 6  , that i s ,  |u| _< n ' _3  We can assume  n > 6  uniformly, by i t s p a r t i a l  1  i n t e r v a l , say,  ^ = 6  n  .  I(x,n) n  t o depend on  &  n  < 6  '  .  Then  consists of f i v e i n -  6  n  .  , so t h a t  (-»,-6) , (-6,-6 ) , (- >& ) > (  could be allowed n  P  • "5 ' nu^ t o some f i n i t e  our r e g i o n o f i n t e g r a t i o n f o r  If  a.^ "" )  Y  Denote t h i s power s e r i e s by  E r c. . ( n u Y i=0 j=0 1  where the c. .  We  r=3  double power s e r i e s i n the two arguments convergent  .  consider the s a d d l e p o i n t 1 approximation.  " r=3  of  such  ,  * Re{h(u)}/u| _< -K  P >0  6-^ > 0  n  n  6  > ) 6  n  a  n  , we could  d  ( ' ) • 6  ro  take  and thereby o b t a i n o n l y three s u b - i n t e r v a l s o f i n t e -  g r a t i o n and a s i m p l i f i c a t i o n o f the proof.  However, s i n c e  1  •26. 6  n  - 0  ,' |M(c+iu)/M(c)| p < 1  would not then be u n i f o r m l y boun-  ded  by  f o r 6 <_ u < eo and a l l  the  r e s u l t of Lemma 2 . 1 . 1 i s t o h o l d .  LEMMA 2 . 1 . 3 .  ;  j "  6  n  -6  n h  u  (2.1.8)  ( ) du = 0 ( n " )  n h  u  M  n integer  2  Then, p n f i  e  2 n ' r  •  »  .  6 ) > pn 6n  n  n  j  ~P  e  5n'j  p n 6  ' (n > l )  - 6n j• 2  n  f™  J  n n  > p(u -  2 ' du < e  n u  ,  6  -p(u-S')  e  (u -  2 5  6  <  .2  -  pn(u  M  F o r u e (6 ,*)  PROOF.  Hence,  g(u) e c  f o r 'each p o s i t i v e  .  , as i t must i f  .Using the same n o t a t i o n as above,  du + f  g(u) e < >  n  e  -P  n u  du  n  _ i du = p .  n J  6  -pnu  e  2  -pn / . du = .O^e / J 1  5  y  From t h i s and Lemma 2 . 1 .  ^n  f  g(u) e ( > du J "  6 6  < J  nh  u  +  n  :  :|:g(u)||e  nReh  - "  5 n  g(u) e ^ n h  6  e" n  p n u  ( ) | d u + f" u  du + f  -6.  -  n  6  e"  du  s  |g(u)||e  n  n < f  u )  p n U  du  n R e h  ( )|du u  27. = 0(e  p  . )  n  (.2.1.9)  (n > 6 • ) ?  = 0(n- ) M  M > 0  f o r any i n t e g e r  and the c o n c l u s i o n f o l l o w s .  COROLLARY 2 . 1 . 3 .  f  e6  p n u  .  u  2  u  u  M  n l /  M  e  If  -|pnu  some constant  K  = 0(e- ) = 0 ( e ^ n u  replacing  p .with J  u _> 6  2  < M U  e  n  (2.1.10)  °;  , we  -|pu  2  independent 2 p n u  )  ^p  .  have  < K  of  in  p n u  1/3 du = 0 ( e -  c l e a r why  c  p n  )  i s chosen t o be a saddleI f this.were not  would i n c l u d e a l i n e a r term i n  U  , and  i t s p a r t i a l sum  J  leads to  The p r o o f of Lemma 2 . 1 . 3  6  n  = n  -1  will fail  r a t h e r than  n  since i n t h i s  -1/3 case  2 n 6^  0  P  E T, c'-- (nu) u . Restricting nu to 1=0 j=0 ^ i n t e r v a l i n order to approximate P u n i f o r m l y by  would become a finite  so  ( 2 . 1 . 1 0 ) f o l l o w s by  p o i n t , t h a t i s , a r o o t of equation ( 1 . 4 . 6 ) . h(u)  1)  (n >  n' . and  Equation  2 e-  I t i s now  the case,  ,  n  PROOF.  for  du = , 0 ^ e - * P  M  M > 0  For each i n t e g e r  r a t h e r than » as n - co In the remaining i n t e r v a l ,  (""^n^n^  3  P  is  l  approximated by it's p a r t i a l sums. A  28.  For any p o s i t i v e i n t e g e r  we w r i t e  P  (nu ,u) = 5  A  r • c.. ( m r ) i>0,j>0 3  1  u'  1  .  J  J  i+j<A  LEMMA 2.1.4.  If  |u| < 6  P(nu ,u) - P ( n u , u ) = 0 [ ( n u ) 3  3  5  A + 1  A  uniformly with respect t o u  PROOF. £ d iri>0,n>0  z y m  ,  n  ] + 0[u  and n  A + 1  ]  (2.1.11),  (but not n e c e s s a r i l y A),  Suppose an a r b i t r a r y power s e r i e s , , converges f o r  n  |z| < R  , |y| < S  .  ,  11111  Since the terms of a convergent power s e r i e s are bounded, d  mn  °( "  =  R  m s  " )  •  n  ^ mn m_>0,n>_0 E  d  m  n  Z  V  Then, i f  = °(  m+n>A  |z| < R  and  m  K  n  &  m'+n>A /  r (||| + |||) ) k  k=A+l  K  b  = 0((||| + | | | )  r > 1  ,  * i|| ||| ) m>0,n>0  = o(  Since, i n g e n e r a l ,  |y| < S  |a +•b| <. 2  , i t follows that  r  r _ 1  A + 1  (|a|  r  )  .  + |b| ) f o r a l l r  £ , mn ^'V = 0( | z | m>0,n>0 d  m+n>A  1  A + 1  ) +. ( |y| 0  A + 1  )  29. Equation ( 2 . 1 . 1 1 )  i s an iriiraediate consequence.  We now come t o our main theorems i n which we s h a l l prove, w i t h the a i d o f the p r e c e d i n g lemmas, t h a t the expansions given i n (1.4.18) and ( 1 . 4 . 2 5 ) are asymptotic.  THEOREM 2 . 1 . 1 .  L e t d. = ( - a ) " l ^ 2'  > ( i = 0 , 1 , 2,. . . )  xTfrn+i+i)  I  H  9  m  =  Q  c . (-a ) m,2i-m 2' 0  m  v  m  9  , where  (2) a  2 ~ "  and  p.  ( )/  K  c  2  * the  t  c m n  }  are d e f i n e d  i n equation  denotes the- gamma f u n c t i o n , that i s , (2n).' .  r(n+*) = nTi """""  i  1  25  Then,, i f •, J M  •  < .» f o r some  J  ,  1  PROOF. < 0  j _> 1  n  j  i s ; a n asymptotic  2  (2.1.12).  • n h (u) » . i ; g(-u) e ' -o Y. d. n " * " (n—) v - : . . , . . i=o a .  CO  a  (2.1.13)  1  expansion.  From (2.1.10) we o b t a i n , r e c a l l i n g t h a t  ,  /  ,-8 . ' ,» i ^ na u . !a n {J. + j . } P (nu^,u)e ^ du = o ( n e •j -eo 6 2  f  (2.1.7),  n  1 / / 5  ?  n  2  A  A  n  f o r any f i x e d  d  (n-«) (2.1.14)  A  Hence, combining the above and t h e . r e s u l t s of Lemmas 2.1.1, 2.1. 2, 2.1.3, and 2.1.4,  30.  n a u2  :/•: « nh (u) f" i|J g(u)e " J _ P (nu ,u)e 1  d  u  w  — 00  •r^-'S  I If.-6  2  A  nh^(u) du | .+ • ,|.{[-6  ^/}s(u)e  < I{J  +  p  5  n  n  2 J } P ( n u , u ) e na„u du|  +  3  dul .6 . nh,(u)' + J }g(u)e ^Vdu| n  2  A  6 + |J [ g ( u ) e n  n h  ( ^ u  - P (nu ,u)e 3  u n a 2 U  A  2  ]du|  ~ n 5  - 0(n" ) + M  1  na~u 2 CO  /  3  n )  2  A  2  o(J. e (|nu3|A+1 + |ul .oo  +  0  *a n (e*  —  A+1  (2.1.15),  Now l e t us consider >  -tx  r—  e.  1 -  i n t e g r a l s o f the type^  k  x  stituting'.  (n-~)  )du)  dx  ,  tx = y r  where  R e ( t ) > 0 ... F o r even  -m-|  2  x  dx  =  t  =  ,  -  -  r ( m + J )  m  -  i  !  ^  ^  ( : m = 0  ,l,...)  mJ 2  The f o l l o x v i n g estimate i s v a l i d f o r both odd and even 00  |  2 |ex jdx = 0([Re(t)]-*( " ) t x  , sub-  ,•we o b t a i n  - t x 2m.  e  k  k  k 4  1 )  (2.1.16).  k  :  .(2.1.17)  — oo  using (2.1.17), we see t h a t the l a s t term i n (2.1.15) i s  31. 0(n"-  )  A _ 1  .  T h e r e f o r e , combining ( 2 . 1 . 1 5 ) , (2.1.16) and  (2.1.17),  •eo  m>0,k_>0 'm+kCA  x . (  )-*(3m+k+l) p ( - | [  a  where  is 0  or  even, r e s p e c t i v e l y .  1 As  we o b t a i n an asymptotic » . -,M  g(u)e  nh.,(u)  i ] ) + o(n-* - ) + 0(n" ) A  5 m + k +  1  depending on whether A  and M  (n-) ,  M  i  i s odd or  are completely  arbitrary, !  series » i . E d.n * i=0  du ~  (n-»)  1  -co .  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 . • THEOREM 2.1.2. L e t e r  =  » „j K ( j ) E • -if E j=0 ' k=3j J  x c, ({a } ) u  , where  k  a  = K  , K ( j ) •'= 00 ( j > 0)  K(0) = 0  ( r )  (c)i ri  r  (r=3,4,...)  ,  /'and - c ( { a } ) (k=0,3,4,. .. ) k  r  are a p p r o p r i a t e l y d e f i n e d constants depending o n l y on . ( a ) (r=3,4,...) r  as before  .  (see equation g(u)e  .C/2T x n  j  Then, w i t h  ^  k  /  2  du ~  Q (c,/C2lva^)/j.' k  z  defined  (1.4.20)),  - »  _  \ ( ) (k=0,l,... )  E E 'c j=0 k=3J '  \) (  _2a )  (n—)  r / 2  2  (2.1.17)  '32. ••is-, an^asymptotic  expansion.  PROOF.  The proof i s almost i d e n t i c a l t o the proof  of Theorem 2.1.1 .  and w i l l o n l y be o u t l i n e d b r i e f l y here.  ' In t h i s case, the f a c t o r  i n t e g r a n d of  (1.4.19) i s expanded 3  argument  (nu^) .  |hu | _< 1  Lemma 2.1.3 2.1.3  2,  au ~ * . r  P'  3  i  n  t  h  e  as a power.series i n the  2.1.1,  , that i s ,  3  to approximate  5  3  T h i s s e r i e s i s denoted by  As i n the p r o o f of Theorem •interval  . nu e  mr  P'(mr)  i s r e s t r i c t e d t o the = -6  |u| _< rT ^ 1  , i n order  u n i f o r m l y by i t s p a r t i a l sums  P^  can be a p p l i e d without a l t e r a t i o n , w h i l e C o r o l l a r y  i s obviously v a l i d with  2  2  e" • replacing e " , c+iu since the l a t t e r . i s l e s s i n modulus than the former everywhere. p n u  p n u  In the same manner as we obtained f i n d that, f o r lul < 6 n 3  and  we  ,  (P' - P ' ) ( n u ) = 0 ( [ n u ] A  uniformly i n u  (2.1.11),  n  3  .  A + 1  )  Then /  2 \ [,  g(u)e  du = I  1  -oo  +  0(n-«)  p;(nu ,u)e 3  2  -a,  +  0{f  C+XU  ,  e^lnu'l^du}  (2.1.18), Writing  p;(nu ) = 3  E  j=0  ( E J  '  r=3  a u ) ' r  J  33.  =  'A ^  3=0  j K(j) k ii_ £ c, ({a }) u J  '  - k=3j-  ,  J  _ k / 2  , equation ( 2 . 1 . 1 8 ) . becomes  •  ;  nh  c  n '  K  n  (u)  A  K(j)  ,  ,-r  f  + 0(n ) + 0(n"^ " )  Q (c,/T2naJ) j2lT/y.  _ M  k  A  1  (2.1.19).  (n-eo)  Again, as  M  and  above, and hence ( 1 . 4 . 2 5 ) ,  A  are e n t i r e l y a r b i t r a r y ,  i s an asymptotic  the  series.  T h i s completes the proof of the asymptotic nature of the s a d d l e p o i n t 1 and 2 approximations. a case not i n c l u d e d i n Theorems 2.1.1 2.2  We now  and  2.1.2,  The L a 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 p r e c e d i n g argument f a i l s ; i n t e g r a l ( 2 . 1 . 6 ) cannot  the t a i l s of the  be ignored, s i n c e f o r a d i s t r i b u t i o n  h a v i n g i t s mass concentrated at p o i n t s  h  u n i t s apart, the  c h a r a c t e r i s t i c f u n c t i o n i s p e r i o d i c of p e r i o d with  discuss b r i e f l y  |cp(X)| = 1  |c?(s)| <,1  and  Daniels [ 5 ] ,  for  X = 2Tr/h  0 < s < X  ,  .  i n h i s work i n v o l v i n g the d e n s i t y func-  t i o n , avoids t h i s c o m p l i c a t i o n because he i s d e a l i n g with d e n s i t i e s i n s t e a d of d i s t r i b u t i o n s . of i n t e g r a t i o n stops at present.  c + ITT  In t h a t case, the path  , and no t a i l r e g i o n s are  Using t h i s f a c t , an approximation 1  to the  distribu-  t i o n f u n c t i o n could be obtained by n u m e r i c a l l y i n t e g r a t i n g the d e n s i t y f u n c t i o n . Feller approximant  F*  [7] i n t r o d u c e s the concept to  F  n  of a p o l y g o n a l  , where, more g e n e r a l l y ,  G  #  i s the .  c o n v o l u t i o n of h/2)  and-  h  G  v/ith the uniform d i s t r i b u t i o n on  i s the span of  G  .  He then shows that the  f i r s t two terms of (1.2.2) approximate of magnitude> o(n"^) of  F  n  .  , the e r r o r i s  |-[F (x) + F (x-) ] n  .  (-h/2,  w i t h an e r r o r  T h i s means that at the l a t t i c e p o i n t s o(n~ )  when  2  F ( ) x  r e p l a c e d by  i s  n  However, f o r h i g h e r order  of the type (1.2.2) the a d d i t o n a l assumption  lim sup |s|-«  expansions  that  |cp(s) | < 1  (2.2.1)  i s necessary, a c o n d i t i o n 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 c o n s i d e r a b l e 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. order of magnitude of the e r r o r i n approximating  F  The  by a  s e r i e s of the Edgeworth type depends on the a r i t h m e t i c a l nature of the set of p o s s i b l e v a l u e s of the random v a r i a b l e Even i f a l l the moments of  F  are f i n i t e ,  X.  i n the case of  !  d i s c r e t e d i s t r i b u t i o n s i t i s necessary t o supplement the expansion '  (1.2.1) w i t h d i s c o n t i n u o u s terms. However, although i t i s i m p o s s i b l e t o  approximate  s u c h " d i s c r e t e d i s t r i b u t i o n . f u n c t i o n s w i t h continuous f u n c t i o n s to an accuracy of w i t h i n .one-half of t h e i r maximum jump, l o c a l l i m i t - t h e o r e m s f o r approximating continuity exist  F  n  (see Gnedenko-Kolmogorov [ 8 ] ) .  the' f o l l o w i n g p r o p o s i t i o n of Esseen analogous  to the Edgeworth expansion  continuous case.  at i t s p o i n t s of d i s -  (see [ 8 ] ,  We  reproduce  p. 241) which i s  (1.2.1) i n the a b s o l u t e l y  35. X^ ( 1 = 1 , 2 , . . . )  Suppose the random v a r i a b l e s only take on the values  = a + sh (s = 0 , + 1 , + 2 , . . . )  x o  where  h  can ,  """"  i s the maximum span of the d i s t r i b u t i o n  P  ..  The  random v a r i a b l e •n k=l  oM  n  K  can o n l y take on the v a l u e s  y  where  p =  ns  r, i p  =  i  (  h  S  " P)/(°^ ) > n  H  and p^ = P(X = a+ih)  . Let  k  l=-eo  P (s) n  = P(Y n = y ^ ) .  THEOREM 2 . 2 . 1 .  (Esseen).  I f the i d e n t i c a l l y  t r i b u t e d random v a r i a b l e s '. -X^,...,X  are independent k ( k >_• 3 ) ,  f i n i t e - , absolute moments of the order  ;  'n< ) P  s  r ^ . ^  TJ^^ns^  =  . + 0(n"(  k _ l ) / 2  ±  /  2  and have  then.  \(n(Y  ) '•  dis-  n  s  ))  (n—).  Here, R  l( ( ns n  y  , ,  D  ) }  =  " 37 *  R ( ( y ) ) = TI n  2  and the R^(n(y)) from the expansion  ( i > 2)  (y)  (k),• .  k  X u  0 )  n  (y)  >  10 , 2 +  sr 3 x  (6), n  x  ( ) y  '  a r e obtained i n a s i m i l a r manner  ( 1 . 2 . 2 ) by r e p l a c i n g  N^^(y)  by  n^^(y)  36-  PROOF.  See Gnedenko-Kolmogorov [ 8 ] , p. 241.  To o b t a i n the values o f the f u n c t i o n p o i n t s of d i s c o n t i n u i t y  y  (y ) = F (y , + 0) n ns n n,s-l u  ;  V J  =  n  a t the  now only r e q u i r e s a summation  procedure,  F  F  S r<s  ;  P (r)  /  37.  CHAPTER 3  y. ;,. ^  ' •;• .'. . "' COMPUTATIONS . T  o  judge the q u a l i t y . o f t h e  i n the case, of s m a l l • n  .  :  saddlepoint'approximations  s e v e r a l t e s t cases were  Numerical r e s u l t s were obtained  considered.  i n each case f o r the sake of  comparison with the Edgeworth and Cramer  approximations.  These r e s u l t s were obtained f o r values of the argument, s e l e c t e d t o r e p r e s e n t the e n t i r e admissable and f o r values of  n  x  ,  range of values  between 1 and 40 i n c l u s i v e .  F o r bre-  v i t y , only a few r e p r e s e n t a t i v e 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 d e p i c t e d .  *  I t w i l l be noted expansion  c  that whereas the s a d d l e p o i n t 2  g i v e s u n i f o r m l y b e t t e r r e s u l t s than the other  approximations, when  .  the Edgeworth s e r i e s ,  i s close to  0  (1.2.2),  , as we would expect  of the d i s c u s s i o n i n s e c t i o n 1.5.  three  i s q u i t e good on the b a s i s  However, when  x  assumes  v a l u e s i n the e x t r e m i t i e s of i t s range, the s a d d l e p o i n t method gives s u b s t a n t i a l l y b e t t e r r e s u l t s . 3-1  Remarks on the Tables. When  e = 0  , the s a d d l e p o i n t 1 expansion  does  not e x i s t , and f o r programming purposes, the- Edgeworth approximation  i s printed i n i t s place.  When  F (x). i s n e a r l y 1, exponents i n the c a l c u l a -  t i o n of the Cramer formula are e x c e s s i v e l y l a r g e f o r the comp u t e r , and the value  F (x) = 1  i s assumed.  38. M u l t i p l e e n t r i e s under the headings c e s s i v e approximations  r e p r e s e n t suc-  obtained by adding, at each stage,  more term to the approximations.  one  They are i n c l u d e d to f a c i -  l i t a t e a comparison of the r a t e s of apparent  convergence  of the v a r i o u s s e r i e s . The r e s u l t s are p r i n t e d i n e x p o n e n t i a l format, a s e r i e s of d i g i t s ,  say  O.n-^. ..n^  , f o l l o w e d by  and  "D +. m"  +m r e p r e s e n t s the number  O.n-^. ..n^ x 10—  o c c a s i o n a l l y r e p l a c e s the l e t t e r In order t o observe  D  ,  The  in this  E  format.  the e f f e c t of the l o c a t i o n of  the s a d d l e p o i n t on the v a r i o u s approximations, c  letter  the value of  and the accuracy to which i t i s c a l c u l a t e d i s given. + or - 5"  Hence, " s a d d l e p o i n t =  means  c e (c^-6,c^+fi)  When i t i s a v a i l a b l e from e x i s t i n g t a b l e s , c o r r e c t value of  P ( ) x  n  l S  given f o r comparison.  the  Prom  these cases i t appears t h a t the l a s t e n t r y f o r the saddlep o i n t 2 approximation  j •  i n each case i s accurate a t l e a s t i n  the d i g i t s where i t . and the next to l a s t e n t r y agree.  In  the .remaining cases,, judgement on the q u a l i t y of the v a r i o u s approximations /available.  must be w i t h h e l d u n t i l exact v a l u e s become 7  I f .the l a s t s a d d l e p o i n t 2 e n t r y i s accurate to  the extent j u s t . d e s c r i b e d , as seems l i k e l y to be the , - an examination approximation  case,  of the t a b l e s i n d i c a t e s t h a t t h i s method of  g i v e s r e s u l t s of the same comparatively good  q u a l i t y as i t d i d 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 V a r i a b l e s . X  Let  = |Y l  ±  Y  where  i  ±  ( i = 1,2,...)  independent, standard normal v a r i a b l e s . of  are  The d e n s i t y f u n c t i o n  i s given by ET f (x) =  2  VTT  e"  /P  x 7  !  ,  0  x 2 0 x < 0  ,  and the moment g e n e r a t i n g f u n c t i o n i s  M(t) = 26*  /  N(t)  2  1  where '  ,  v /2 2  N(t) =  J  e  y  /  dy  — eo  The cumulants are given by Rubin and Zidek  ^1 a  2  =  a  3  a  4  as  i Z 0-79788 45608 03  = a  = 1-a  2  2  ~ 0.36338 02276 32  a (a -a ) ~ 0.21801 36l4l 2  a  [13]  1  2  45  = 2 a ( 2 - 3 a ) ~ 0.11477 06820 54 2  2  (3-20a +24a^) ~ - 0.00443 76884 6262 2  The s a d d l e p o i n t  c  i s the r o o t of the equation  n(z )/N(z.) + z ='x/n  which/can be solved n u m e r i c a l l y u s i n g Newton's Method.  ,  40. In order t o compute the c o r r e c t value of equation ( 1 . 4 . 4 ) was  F (x)  inverted numerically i n [13];  e n t a i l e d the e v a l u a t i o n of  This  M(z), and hence, of  complex v a l u e s of i t s argument.  ,  N(z)  , for  Since use of i t s T a y l o r ex-  pansion r e s u l t s i n u n c o n t r o l l a b l e round-off e r r o r , an ac- • curate method u s i n g continued f r a c t i o n s was computation W(p)  employed.  (see equation ( 1 . 4 . 2 1 ) ) a l s o r e q u i r e s  of . Q (p) Q  , and f o r t h i s reason, a d e t a i l e d account  thod devised i n [13]  The  of the  me-  i s given i n Appendix A.  The d e r i v a t i v e s of the cumulant g e n e r a t i n g f u n c t i o n evaluated at  c  are  K-^ = x/n = x K  2  _ .. _2 = ex + 1 - x  K^ = c x + c ( l - 3 x ) - x ( l - 2 x ) 2  2  2  K  4  = c \ + c ( l - 7 x ) - cx(5-12x )' + x ( 4 - 6 x )  K  5  = c x+c (l-153c ) -  2  4  2  2  3  2  K  2  4  2  2  + 243c ) 4  c%(42-l80x )  = c^x + c ^ ( l - 3 1 x ) 2  6  2  c x(l6-50x )  - c(3-35x +6dx ) + x(3-20x 2  2  2  - c (13-191x +3903c ')'+ c x ( 4 l - 2 7 0 x + 3 6 0 x ) 2  2  i|  2  - x (28-120x +120>c ) 2  4  2  l1  .  n From now on, F (x) w i l l denote P( £ X. < x) . Then, f o r i=l ~~ the s a d d l e p o i n t 2 method, the f i r s t term approximation to n  1  f  Table 1(a)  11  CHI RANDOM VARIABLES (=ABS'( V) .WHERE Y = NORMAL ? ME AN = 0 t VAR . = 1 ) N = IO X SADDLEPOINT F( X )  = 3.6 00 00 = -0 . 2 1192 3 8 ' 0 1  =  EDGEV/ORTH  0 . 1080667090- 01 0 . 4409401340- 02 0 . 405517680D- •02 0 . 404377742D- 02  0.410254 E- 02  CRAMER  0 .4046812430-02  EDGEWORTH  CRAMER  0.366E-13  SADDLEPOINT  1  0 .4452880830-02 0.3978365120-02 0 . 4 174711670-0 2  X 7.00000 SADOL EPOINT = -0.2939604 ,'E 00 F( X) = -.. ;  + 0R-  = X f . .' SADDLEPOINT = F (X ) =  EOGEWOR. IH  0. 0. 0. 0.  8979775370 00 8922496760 00 8929280630 . 00 8 92 7336.62D_ 00.  0.38334 64480 -02 0.^106579370 - 0 2 0.409961609D -02 0.4 102644 590 -02 0.4102950790 -02  !  ....SADDLEPOINT  1  SADDLEPOINT  2 00 00 00 00  0.3015973490 0.318787413D 0.31871.34320 0.3183.5 3 2480 . 0 . 3 18.353325D  CO  0 0 . +0R- 0.6I9E-14  0.89276430 E 00 CRAMER  0 .8883722670 00  X SADDLEPOINT .. F.(X.) EDGEWORTH  10.40000. 0.5623426;'E  2  + 0R- 0.5 55E-16  0 .3014527160 000.6766890570 00 0 . 3 0 33036570 00 - 0 . 1.67467490D 01 0 . 31730 92 810;. •00 " o. 3 18:6 01 1.58 0'' oo. • • 0.2359028000 0'2 •• o: 3.1 88 1'48570 . 00  V•'.  SADDLEPOINT  SADDLEPOINT  1 •  0.8 45 7 06 7.380. 00 0.9589.380730 00 0.7223566340 00  SADDLEPOINT  2  0.83803012 30 0.8 9250332 ID. 0.892749136D 0.8 92758476D 0.8927634590  00 00 00 00 00  •= 13.90000 = O . U 5635l',E OL +QR- 0.222E-15 _=... 0.99729399 E. 00 . CRAMER  0.996790.3210 00 0.999052373D 00 0.9975984990 00 0.9 9 7.199 8 5.10 . 00 0.9973176900 00  SADDLEPOINT  1  0.9 9 696 46 2 50 0 0 0.997404934D 00 0.997230346.0 00  SADDLEPOINT 0.9972253000 0.9 9728188 50 0.9972943640 0.9972937810 0.9972939980  2 00 00 00 00 00  A2. i  CHI RANDOM VARIABLES {= A 8 S ( Y ) »WHERE Y = NORMAL,MEAN = 0» V AR . = 1 ) N •= 40 •  X =*""l8. 7 59 99 SADDLEPOINT = - 0. 1 3 079 50' E 01 :  F( X)  •=  EDGEWORTH '  0 . 6 2 7 4  CRAMER  0.2796856250-03 -0.1671882140-04 0.5 64 75 6959 0-04 0.63725764.30-04  .  0 ."6323368520-04'  .  SADDLEPOINT  .CRAMER  0 .302715069D- 01' 0 .257362.3 94 0- 01 0 .254511868D- 01 0 .254154082D- 01  0.2465491280-01  . .. - - .  - -  0.6128094370-04 0.628270354D-04 0 .6 2743 I9 54D-04 0.627446602D-04 0.6274432780-04  SADDLEPOINT 1 0.298779195D-01 0.2 27366469D-01 0.2 818401640-01  SADDLEPOINT 2 • 0. -2472592890-01 0 .2542728.390-01 0 .2 54 1.0 1692.D-•01 0.254113 0 69D- 01 0.2541123900-•01  39.00000 0.42 83156- E 00 +0R-. 0.416E-16 0.96412521 E 00 CRAMER  0 .9684337340 00 0 .963 367 5460 00 0 .964158 50 ID 00 0 •.3641.2J.600D_ 00  SADDLEPOINT 2  +QR- 0.694E-15  -•  X = SADDLEPOINT = F{X) = EDGEWORTH  1.  0.6 5 9743 3 76D-04 0.621543542D-04 0.6292393050-04  .' X; ';. . •'. = 24.75999 . '';S AODL E PC I NT •= -0 . 5843487: '.E 00 F(X) - = 0. 2541083 E - 0 1 ., i;:EDGEypriTfj.._y  +0R- 0 . 1 U E - 1 4  E - 0 4  0.963448929D 00 .•'  SADDLEPOINT 1  SADDLEPOINT 2  0.9556373200 00 0.,.9702510900 00 0.95638 55400 00  0.9633715030 0.9640933810 0.96V124178D 0.9641242910 0.9641244540  00 00 00 00 00  45.00000 X SADDLEPOINT = . 0.72 39 73 4; '.E 00 +0R- O.OOOE 00 F( X) = 0.999317.10. E 00 EDGEWORTH 0 .9997004530 0 .99938S199D 0..9 9931.1 19.40 0 .9993169090  CRAMER  oo • 0.999297909D 00  00 .00 00  SADDLEPOINT 1; .0.9992547.280 00 0.9993332-770 00 ..0.9.993 100 14D 00  SADDLEPOINT 2 0 .999 307 3420 0 . 9 99.3 161.800 0.999317116D 0.9993171000 0.999317104D  00 00 00 00 00  43F (x) n  is  . , >> F ( x )  sign(c)) -  n  where •"'•1 . V  Q ( p)  i s given by  Q  e  -  x  c  +  n  K  (  c  )  3-3  Q  ,  (1.4.21).  The r e s 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 v a l u e s of those  Q (p)A/^  F ( x ) 'given are n  computed by Rubin and Zidek [13]. The E x p o n e n t i a l P r o b a b i l i t y Law. The p r o b a b i l i t y d e n s i t y f u n c t i o n of the e x p o n e n t i a l  d i s t r i b u t i o n i s given by  f(x)  =  Xe  -Xx  , x > 0 ,• otherwise  0  where  X > 0  .  The moment g e n e r a t i n g f u n c t i o n i s  M(t) = x/(X-t)  (Re[t) < X)  ,  from which we o b t a i n the cumulants  a  ±  =  K (  X  \ O )  = i.'/X  1  S o l v i n g the s a d d l e p o i n t equation  c = X - (x/n)  ( i = 1,2,...)  yields  .  44. and  hence,  K ^ ^ ( c ) = r< X ( x / n ) r  Difficulties approximation. the  argument  example, when  '  x  (1.3-4)  , series X = 1"  w  ^  ,  (1.3-4)  1  = j  -  , vj/Jn  be' evaluated u s i n g  . 1 2 +  1 -  ^Z-  |z| > 1  3 „,  -g  Z  .  results.  ,  x(w/7n)  We overcame t h i s  difficulty  0  c  indicates  r e c t i n the f i r s t .-significant  x = 4  and  that  f o r moderate v a l u e s  Hence, the Edgeworth s e r i e s o f t e n y i e l d e d F o r example, when  cannot  (1.3-3).  can be c o n s i d e r a b l y d i f f e r e n t from x  . . .  , and  The form of the e x p r e s s i o n f o r  of  +  Thus, i f x > 2 0  .  = (x-nfi )/(no) > 1  by i n v e r t i n g d i r e c t l y the equation  it  and  becomes  ^ Z  (1.3-4).  X  does not converge. For  1  which does not converge f o r "" n = 1 0  .  are encountered i n a p p l y i n g the Cramer  F o r c e r t a i n choices of the parameter  X(z)  and  (r = 1 , 2 , . . . )  r  n = 15  inaccurate  i t i s incor-  figure.  R e s u l t s 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 I A L RANDOM V A R I A B L E ' MEAN= 1 VARIANGE=1  15  SADDLEPOINT =  =  F(X )  EDGEWORTH 0 .99993 19550 0.9998733900 0 .999634452D c.9994320490 0.9994370310  31 .OOOOO 0 . 5 1 6 1 2 9 0 ' .E 00  CRAMER 00 00 00 00 00  + 0 R - O.COOE 00  0. 99948'^  0 . 9 9 7 2 1 1 0 8 7 0 00  S A D D L E P O I N T 1'  S A O D L E P G FNT 2  0 . 9 9 9 4 1 7 6 5 5 D 00 0 . 9 9 9 4 9 1 4 0 8 0 00 0.9994697970 00  0 . 9 9 9 4 4 7 0 5 2 0 00 0.9 9 9 4 7 57 2 00 00 0 . 9 9 9 4 7 6 2 3 5 0 00 0 . 9 9 9 4 7 6 3 7 3 0 00 0 . 9 9 9 4 7 . 6 3 2 9 0 00  11.00000 SAOOL EPOINT '= - 0 . 3 o 3 6 3 6 3 E CO  X  .EOGEWORTH .  0. 1 5 0 8 4 9 0 5 8 0 0 . 1 4 9 50 70 06 0 0 . 1466.16392D 0. 1460833740 0 . 145980.7790  +0R- O.OOOE  00  • • = 0.14596  F (X )  SADDLEPOINT 1  .CRAMER 00 CO CO 00 00  0 .1339353650 00  0 . 2 0 1 1 8 4 7 3 4 0 00 0,6 1 7 5 2 53 640-0.1 0 . 3 8 2 3 7 6 3 4 7 0 00  5.75000 S A D D L E P O I N T •= - 0 . 1 6 0 3 6 9 5 - E 01 +0R- O.OOOE 0 0 F(X) 0.93 • E-03 X  EDGEWORTH 0.8462344 170-02 -0.86121708.20-03 - 0 . 3 1 8 4 2 6 7.690-03 0 .'59.6'495454D-03 C. 9 0 1 4 1 1 9 6 7 0 - 0 3  CRAMER 0.9447172850-03  X  SADDLEPOINT F {X )  EDGEWORTH 0 .2254348950-02 -0 . 2 0 4 4 0 1 6 9 0 0 - 0 2 -0 . 3 9 3 2 7 6 2 0 5 0 - 0 3 0.2349010740-03 0 ,2 1 2 7 0 6 2 1 4 D - 0 3  2 .  0. 1 3 3 9 2 7 2 0 2 0 0. 1 4 5 4 3 3 2 7 9 0 0 .-145 89 55 5 6 0 0. 1 4 5 9 5 2 6 5 6 0 0.1459570810  00 00 00 00 00  •  SADDLEPOINT 1  SADDLEPOINT 2  0 .98556825 10-03 0 .913860382D-03 0.9347567530-03  0.86498 30210-03 0 . ^ 2 6 1 8 3 0 780-0.3 0^9279023940-03 0.928.3587020-03 0.9284334050-03  . 4.00000 - 0 . 2 7 4 9 9 9 9 . £ 01  0.2  •SADDLEPOINT  +0R- O.OOOE 0 0  '" E-04  CRAMER . 2980789600-04  SADDLEPOINT 1  SADDLEPOINT 2  0.2062211930-04 0. 1 9 8 2 5 8 2 9 0 0 - 0 4 0. 199 59 51.8 3 0 - 0 4  0. 1 3 6 8 2 9 7 2 4 0 - 0 4 ' 0, 1 9 9 0 0 2 2 4 5 0 - 0 4 0. 1 9 9 2 1 4 8 0 7 0 - 0 4 0 , 1993001850-04 0. 1 9 9 3 1 5 9 1 3 0 - 0 4  i a b _ l e _ j n i ( . b _ ^  EXPONENTIAL RANDOM V A R I A B L E ME AN = 1 VARI /- CE=l f,  N = 40 X = 15.50000 SADDLEPOINT •= - 0 . 1 5 3 0 6 4 4 ;E 01 = F(X ) EDGEWORTH  CRAMER  0 .5357784620-04 -0 . 1 0 8 7 9 5 5 6 1 D - 0 3 0 .6 1 8 0 1 6 9 9 2 0 - 0 4 0 .4954126 660-05 -0 . 6 9 7 7 1 5 1 7 6 0 - 0 5 .  0 .179693256D-06 i  SADDLEPOINT 1  SADDLEPOINT  0 . 1527063360-06 0 . 1484449270-06 0 . 148 9 1 9 9 0 0 0 - 0 6  0 . 1 4 4 0 9 0 1 1 2 0 -06 0.14886^4730 -06 0. 1488 38 31.70 -06 ' 0 . 1 4 8 8 4 7 6 9 5 D-06 0.1488491500 -06  X = 30.00000 SADDLEPOINi T" •= - 0 . 3 3 3 3 3 3 3 E 00 F (X) = . EDGEWORTH  CRAMER  0 .5692336810-01 0 .478 87 2 484 0-01 0 .4638122850-01 0 .4625760940-01 71560.-01.. _ _ p . .4 62.51  0 .4401459090-01  X SADDLEPOINT FIX) EDGEWORTH 0 .7854023250 0 .7911709600 0 .7916667020 0 .79 1 6 1 2 6 7 6 0 0.7916188290 :"' , -  -  0.5589016810-01 0 . 3900 667 99D-01 0.558087329D-01 -  45.00000 0 . 1 1 1 1 1 1 1 E 00  . SADDLEPOINT 2 0 . 4 4 0 1 0 2 4 0 1 0-01 0 . 4 6 1 9 2 7 6 9 1 0 -01 0 . 4 6 2 4 7 1 5 0 8 0-0.1 0.462 52 7 9 4 3 0 -01 0 . 4 6 2 5 3 2 1 2 5 0-01  +0R- O.OOOE 00  =  CRAMER 00 0 .780228869D 00 00 00 ' 00  SADDLEPOINT 1 00  SADDLEPOINT  2  0.621907080D 00 0.7802289270 0 . 1 3 0 3 2 6 2 0 3 0 01 . 0 .7918598.340 -0 . 2 2 3 7 1 0 6 3 7 0 01 0 .7916026090 0.7916187610 0 .7916182510  00 00 00 00 00  - 55.00000 '• 'X ; '• SADDLEPOINT- ^ 0 „ 2727272. E 00 '7+0R- O.OOOE 00 ' F (X ) . - . -  :E,DGE WORTH  :  =  -  2  +0R- O.OOOE 00  SADDLEPOINT 1  * -  +0R- O.OOOE 00  .  0 .991. 1 4 6 9 6 8 0 .0.0 0 .9 8 5 3 0 6 9 C 8 D 00 0 .9.8 5 1330 3SD 00 0 . 9 8 5 3 4 1 2 7 5 D 00 0 .98 5 29 7 5 50D 00  ^CRAMER;.  . SAD.CLEPOINT 1.  0 .9846229.370.: GO,. • 0 . 9 8 2 4 7 9 7 8 2 0 0 0 0 . 9 8 6 7 9 9 0 0 3 D 00 0.9 8 3 9 3 4 3 9 6 0 . 00  SADDLEPOINT A. 9 8 4 6 4 5 4 8 70 0.9853127.530 C. 35301129D 0.9853029960 0.9353028100 Q  2 00 00 00 00 00  rr ro r>~  cr- C J CJ c r- r- r- r-  h-  O' o <?• cr  c  eg ;  c o < 0'. o o  ro  o  o o  1—  o  r—-  -  1  o  Ci.  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CJ  :  i  ;JJ  O UJ  ,—1  ai  1 I c oc. a co c c, c ,  . cr  •4- <r~ o C J c- o CJ-  CJ  CJ  CJ  rv P-J rv •ij .^J ^0 rv rv p. cv rv CO CP ro ro ro rv rv rv c-J a • • • PJ  LL'  ro ro O  P-J  \ Oo C  o  E  cX  ;  '  ; !  ;  <  o ,  ;  ,.  X ~—  a.  T;  H  a  Jr-" Llj Ci!  C LJ  -  1  i—1  o o o. 1 1. 1 1 •  c u". o  LH IP. u > LP; LT'i IP, LP. LP-. LP LPl r ~ l .—• r-^ --i •  o  .—t  o  LP  rCV  rv  «  rv r-vi  «  o  C~'  r-  o  o •  X  X  L/)  U-  • ex. ; O . Jf-  : U.:  P-  r  O  '  UJ  rv oj rv rv CV! 0  9  ooooO i  • o  o CJ o P;  rv  o9  •  C a.  L;._j  c  CJ.  <  0*  •  LP  CJ  rv p.; a- C J  P>!  CJ  — . i r-l  C-  CJ C 9 *  •  o c- CJ- o o  o  o o o C'C :cC J LP. o  0": CV rC J in CJ;  o  LP. CO vO o  ro  co ro cv CJ-  — . i LP. O' CJ-  9  o  o- c--< 9  9  CJ) CJ-  o  OJ c- Co <7'  P0  vO  LT\  r-i  H r<\ OA,  d  o  o:  LL;  '.'i  <.  o rv CJ  r H  p  II n  P!  ro *  o  h1—*  o  o  o  o  o  o-  ST  a  Q-  o  Oc o o  ca •ii  rv pj rv >£ o r—(  o  E  o  X  P4 P J NT  r-1  r-i  CJ  <Ji  X  <l" ij"i  —  ZL.'  HCC  C: .JLiJ  V  G  LL;  oc ooo i  —.  C;  .—i  9  *  o  LL.I  o  --o 0  •  c> O  NT'  r—  CJ  o  c> o c o MC  o  -.0  o  ro ro r<"i rr-, rr-. ro f O r^ rp  5C  o  1  —  X  >-»  •  +  P l PO  o o o  C  LO u-i  PJ  e  C:  o  o  cj.  .—  c, Q ir,  LTi  1-  cc  «  LU  r-i  o C.J 1  cc rv ccLO .—i  a.  r-i  CP;  O  e  •\  1  o  —i CJ CJ  U.J  II  1~ J-_  •—<•  o o  fp| LP.  «  LU  00  <r <3- CJ . ro .—i G' a -cc  _  oo sC  <  9  o o o c.c a  rv  c rv C-  a.  ~r<f •4a-a-  vO vt .—i r-; <-r.  1—1 r\j  a-  CJl  CJ  Goooo  r—  Ol  c oooo  r-J  o c- o  IP; LP; LP- LP  C  IV  II  O  \—  UJ  vO  IT.  cc  <;.  r-l  O - *  G-  xT P.! (V:  -^r .—i  rv  i—i  CT  o  o  £7J  o I c LP.  C: <. r-i o  1  (_• Cd Lb  H rA  r-4  LU  -4" cv c;". •Xao o- rv rP; rv  rv  _J C  o o  1 1  r*~i ro rv co  r~l  o  . LL'  rv cv rv rv (V; 4 a •  o o o o o  o o o o  oa oo o,c c c r—• O  CL.  UJ  *  <.  cr- O'-  >  ooooo  ro  H II II  X  o-- CT-  rv  LU  --0J  O  ooooo i 1 c1 c1 c1 c r-,  i C: w  cr r—i C' r-C" r- <r r-~ co co CO ^0 ro G'- CJ CO ro \j -r..- r~. rv r\! cv • a •  2!  4  o  c  h ~ZL  o O o 1—i 1 1 I oCo CO CC'/  UJ  LU  f\J  CJ CJ  rv  c-  o <; O CV rv r^. cv rv CO P - CO CO rO rv CV: rv rv .. • 0 • • • O o o o o  LL'  o  r/j;  0"  o c O Oo 1 1 1 1 I C: Cc c  o  CJ-  o  o  ooooo c o c c-i  u.. rv cv  -o  <J CJ  Ch  CV  Cr-  r—1 r~1  cv cv CJ—'  "  r<i CO ro m ro ro r<~. CO i-p. CJ  N  CJ- 0 4  CJ-  S  •  •  •  o o Oo o  .Table I I I CM, NORMAL  48.  RANDOM VARIABLES • MEAN- .5 VARIANCE^! N = 36  SADDLEPOINT F (X ) EDGEWORTH  ' 0.00000 • - 0 . 5000000''F 00 +0R- O.OOOE 00 Q.1349898'03 E-02 CRAMER  0 . 1 3 4 '3 97 75 D-02 0.1349397750-02 0.13493 9775 0-02 0 .1349897750-02 0.1349897750-02 0.134989775 0-02 c  SADDLEPOINT 1  SADDLEPOINT 2  0.1477282800-02 0.1313140270-02 0.13678 54450-02  0.13493 93 030-02 0.1349B98030-0? 0 ; 1349898O3D-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 . 6 6 8 0 7 2 0 1 3 E - 0 1  0 .6680733650-•01 0 .6 68 07 33 65 0-•01 0 .6680733650-•01 0 .6680733650-•01 0 .668.07 33.6 5 0•01  '_. ..SADDLEPOINT 1  CRAMER  . EDGEWPR.TH/.  0 .6630733650-01  -  •-  :  0 . 8634506380-01 0 .4 79 69 47 990-01 0.9913692510-01  • -  X 15.00000 SADDLEPOINT •.— - 0 . 8 3 3 3 3 3 1 E- 01" F(XJ ZZ 0. 308537539 EDGEWORTH  CRAMER  0 .3085375400 00 0 . 3035375400 00 0 .3085375400 00 0 .3085375400 00 0 . 3085375400...PC .... 0 . 303 53 7540D 00  S A DDL EPOINT 2  . .  0.6680720130 -01 0 .66807201 3D-01 0 .66807 20130-01 0.6680720130 -0.1 .0 .6680720130 - 0 1  +OR-;:B'-.000E 00  SADDLEPOINT 1 0.704 1.306540 00 -0/. 21 12 39 1 960 01 0.316858794D 02  SADDLEPOINT 2 0 .30? 5375390 0.3085375390 0 . 3085375390 0. 303 53 75390 0 . 3085375390  00 00 00 00 00  X = 24.00000 SADDLEPOINT = 0. 1.666666'-E 00 '+0R- O.OOOE 00 F(x) . • = 0.841344746 EDGEWORTH' 0.8413447370 00 0.8 41344 7 370 00 0.841344737D 00 0.8413447370 00 0.84 1 34.4 7 370 00  CRAMER 0.8*13447370 00  SADDLEPOINT 1  SADDLEPOINT 2  0.7580292750 ..00 0.841 3.447460 00 0. 1 000000000 '' 01 0.3413447460 00 .0. 2 7403 78 260 .00 .0 . 841 3447460 00 0.3413447460 00 0.8413447460 00  .  493.4  The  Normal P r o b a b i l i t y Law.  :  T h i s case i s considered because e x t e n s i v e and . h i g h l y accurate t a b l e s are a v a i l a b l e .  The r e s u l t s  obtained  i n d i c a t e the h i g h accuracy p o s s i b l e with the s a d d l e p o i n t 2 approximation. (1.4.25)  gave an answer which i s c o r r e c t to every  tabulated. and  In a l l cases considered, the f i r s t  term i n figure  However, as the r e s u l t s given i n Tables  Ill(a)  (b) indicate,•' the Edgeworth and Cramer methods g e n e r a l l y  i n c o r r e c t i n the l a s t two 3.5  The Non-Central The  or three f i g u r e s .  ChirSquare P r o b a b i l i t y  d i s t r i b u t i o n f u n c t i o n of the  Law. [X ) i  (i = 1,2,...)  i s given by  F(x|v,\) =  X _> 0  where  T. e ~ ; j=0  (X/2) j.<  / 2  F j x , y + 2j)  J  ,  C  i s termed the n o n - c e n t r a l i t y parameter,  v  i s , t h e number of degrees of freedom, and  F fx,k) = [ 2  | k  P (ik)]'  1  f  X  t  |  k  _  1  e ^ -  dt  o  u  .'  i s the c e n t r a l chi-square d i s t r i b u t i o n v/ith  n  (0 < x < .) ~ degrees of  freedom. The  c h a r a c t e r i s t i c f u n c t i o n of  cp(t) = e x p [ X i t / ( l - 2 i t ) ] ( l - 2 i t )  Using equation  X^  is  _ V / / 2  ( 1 . 4 . 6 ) , we r e a d i l y f i n d t h a t  the s a d d l e p o i n t , i s given by  c,  50.  2  •-  A l s o , the d e r i v a t i v e s of the K  evaluated  at  K^\(c)-=  t = c  cumulant g e n e r a t i n g  are given  (l-2c)" ' 2 ' J  J  - 1  by  (j-l)j  \J7(1-2C)]  +  [v  As a f i r s t approximation to using  function  F (x) n  we  can  '.  write,  (1.4.25),  F (*) -  + sign(c)) +  n  where  p - cvnK^ ^(c) 2  e  -  x  c  +  n  K  ( ) c  +  p  /  2  [ | ( l + s i g n ( c ) ) - N(p)]  .  Numerical r e s u l t s are t a b u l a t e d (b) and  (c).  i n Tables I V ( a ) ,  ,  Table •IV(a)  •  ^  NON-CENTRAL CHI-SQUARE 1 DEGREE OF FREEDOM NON CENTRALITY PARAMETER = 2  X = . 0.5O00O ' SADDLEPOINT = -0.5354101 E . 0 1 . +0R- O.OOOE 00 F { X) = EDGEWORTH  CRAMER  0.2015249080-01 -0.4626224490-03 -0.4282432470-02 -0.350359347D-02 -0.236738097D-02  0.408479631D-03  SADDLEPOINT 1  SADDLEPOINT 2  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 F{X) EDGEWORTH  CRAMER  0.239750076D 00  0.225509436D 00  0 . 2 6 0 2 5 5 0 0 3 D GO  0.260098775D CO 0.2602656350 00' C.2602844530 CQ  SADDLEPOINT = F (X ) EDGEWORTH 0.7602499240 00 0.7807548510 00 0.780911079D  0.7508088640 00  00  0.7810779390 00 0.7810591210 00 X .= SADDLEPOINT = F (X ) = EDGEWORTH 0.9999996280 0.999993707D 0.9999559180 0.999838539D 0..9996826.190  0.4153531 190 00 -0.1803945270 00 0.2696189500 01  SADDLEPOINT 2 0.226020105D 0.258417464D 0.2596917290 0.2602251960 0.2602753060  00 00 00 00 00  +0R- O.OOOE 00  SADDLEPOINT 1 0.5424134760 00 0.1713239310 01 -0.657541799D 01  SADDLEPOINT 2 0.7506725330 00 00  0.781703890D  0.7807042540 00 0.7810869350 00 0.7810573610 00  50.00000 0.2500000 E GO +0R- O.OOOE 00 CRAMER  00 00 00 00 00  SADDLEPOINT 1  20.00000 0.7846480 E-01 CRAMER  +0R- O.OOOE 00  0.100000000D 01  SADDLEPOINT I  SADDLEPOINT  0.999736361D 00 0.999769215D 00 0.9997609390 00  0.9997472650 0.9997612530 0.9997632710 0.9997632280 0.999763223D  2 00 00 00 00 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 F ( X) ;= EDGEWORTH  CRAMER  0.1398500810-03 -0.3643638920-03 0.2044586970-03 0.86.3652947D-04 -0.1594743060-04  0.2933935700-09  SADDLE POI N T F{X) = EDGEWORTH .' 0.2071081480 0.21436946OD 0.2.13783176D 0.2137242390 0.2I3711194D  0»658454280D 0.6 8I.7635 54D. 0,6 810447900 0.681172 4.06 D. 0.68116 1699 0.  SADDLEPOINT 2  0.8276460610-15 0,820674756D-15 0.8205665220-.15  0.7486926930-15 0.8176701410-15 0.819280304D-15 0.820262712D-15 0.8205225520-15  0.333944115D GO -0.7295804330-01 .0.1523558320 01  **  . . .  50.00000 0. 3050665 E- 01 CRAMER  00 , 0 .656302920D 00OO;" 00 ; oo "•• 00  0.9978666380 0.994I95089D 0.993002421D 0.9 934309 84.0 0.9933970090  CRAMER 00 00 00 00 00  0.987962604D 00  +0R- O.OOOE 00  SADDLEPOINT 1 0.962915067D-01 0*6023925130 .01 -0.1072262050 03  • = ' .80.00000 SADDLEPOINT. .- 0'. 1433714 E CO F { X} = EDGEWORTH  +0R- O.OOOE 00  SADDLEPOINT 1  0 .1971838330 00  X = SADDLEPO INT = F(X) = EDGEWORTH  SADDLEPOINT 1  35.00000 •0.82 29 047 E-01 CRAMER  00 00 CO 00 00  +0R- O.OOOE 00  SADDLEPOINT  2  0. 197359615D ' 00 0 .2132536550 00 0.213614325D 00 0.213702119D 00 0.2137072840 00  I  SADDLEPOINT 2 00 00 00 00 00  0.6567731590 0.6814322630 0 .6810769140 0.681164924D 0.681162475D  -+0R- O.OOOE 00  SADDLEPOINT 1  SADDLEPOINT 2  0.992283246D CO 0.993845643D 00 0.993040480D 00  0.9930128950 0.9933702620 0.99.33789250 0.99337.98750 0.9933798700  00 00 00 00 CO  53.  Table.IV(c) NON-CENTRAL CHI-SQUARE 1 DEGREE OF FREEDOM NON CENTRALITY PARAMETER =2  N = 40  SADDLEPOINT  20.00000 •0. 1118 033 E 01  +0R- O.OOOE 00  F('X)  EDGEWORTH  CRAMER  0.2866515030-06 •0. I37847435D-05 0.251425291D-05 •0. 195930593D-05 0.309407186D-06  0.500525922D-13  X = 100.00000 SADDLEPOINT •= -0.5825756 F(X j '= EDGEWORTH 0 . 1 58655263D 0 .1586552630 c .1580584020 0 .157999325D 0 .1579918240  CRAMER 00 00 00 00 00  0 .1505020760 00  SADDLEPOINT 1  SADDLEPOINT 2  0.2775543660-13 0.274821895D-13 0.274975374D-13  0.266054817D-13 0.275465300D-13 0.2749516700-13 0.2749663030-130 .2749738830-13  E-01  SADDLEPOINT .1  SADDLEPOINT  0.229759540D 00 0.322842898D-01 0.568227400D 00  0. 1505807100 0. 1578734720 0. 1579739930 0.1579899980 0.157990658D  = 200.00000 X SADDLEPOINT = 0.1298437 E 00 F (X) •= EDGEWORTH 0 .9999683290 0 .9998746480 0 .9997786290 0 .999746572D 0 .999751-4710  CRAMER 00 00 00 00 00  0.999586453D  00  X = 500.00000 SADDLEPOINT = 0.2790024 F (X ) CRAMER  EDGEWORTH O.IOOOOOGOOD  01  0.100000000D 0.100G000C0D O.IOOCOOOOOD 0.1CC0CCO00D  01 01 01 01  C.100G0C0C00  01  +0R- O.OOOE 00  00 00 00 00 00  +0R- -O.OOOE 00  SADDLEPOINT 1  SADOLEPGINT  0.9997311780 00 0.999757380D 00 0.9997507550 00  0.9997457210 0.9997523140 0.999752574D 0.9997525750 0.9997525750  E 00  2  2 00 00 00 00 CO  +0R- O.OOOE 00  SADDLEPOINT 1  SADDLEPOINT 2  0.100000000D- 01 0.100000000D .0.1 0.1000000000 01  0.1000000C0D 0. 1.000000000 O.IOOCOOOOOD 0.100000000D 0.100000000D  01 01 01 01 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 d e n s i t y  ting function,  respectively,  and moment genera-  of a random v a r i a b l e  u n i f o r m l y over t h e . i n t e r v a l  f(x) =  function  distributed  (a,b) are  (b-a)"  a < x < b  1  0  otherwise  and _i_ \ bt at M(t) = e - e ' (-b-a)t * r /  For  s i m p l i c i t y , we c o n s i d e r the case In g e n e r a l , the cumulants,  a = -b i f they e x i s t , may be  expressed i n terms of the c e n t r a l moments  a  For  = U  ±  a  5  =  a  6  =  U  as  , i = 1,2,3  i  5- "  ^6  {u^}  _  1 0  1  M 3  5wy  M  i  + 2  +  .  ^l^  2 lQ4- (W +6u u +2y ) + 30,^2 5  5  2  1  1  the uniform p r o b a b i l i t y - law, as i s e a s i l y shown,  u  i  •"=:::•  0  bV(i+l)  i = 1,3,5, i = 2,4,6,  3  55, Thus, = -1,3,5,  i  0'  •a g = b / 3 2  2  =  15 b  5  a  6  TH  =  Equation  b  6  b  (1.4.6) becomes  I', , / cb , -cb\// cb -cb , — + b (e + e )/(e - e ) = x/n N  To o b t a i n n u m e r i c a l r e s u l t s , f o r t h i s case, the .last v/as s o l v e d n u m e r i c a l l y f o r and  with i n i t i a l  i t e r a t e = x/n ,  s u c c e s s i v e i t e r a t e s obtained by the Newton technique.  Note. t h a t s i n c e if  c  equation  |K^^(t)|  < b  , a saddlepoint e x i s t s only  |x| < nb • ' 'bt -bt , bt , -bt Let. u = e - e and v = e + e . T  . Then m  the r e l e v a n t d e r i v a t i v e s of the cumulant g e n e r a t i n g f u n c t i o n are  K(  1  ( t ) = - t ' + bv/u. 1  (t) = t " - 4b /u 2  K  (3  K<  K(  4  6  2  2  (t) = - 2 t ~ + 8b v/u 3  (t)  6t"  (t)  -24t  (t)  120t  4  5  3  + 8b (l-3v /u )/u 4  - 5  - 6  2  2  2  - 32b (2-3v /u )v/u 5  -  2  2  3  32b (2-15v /u +15v /u )/u 6  2  2  4  4  2  5d  iTable Y(a) UNIFORM DISTRIBUTION OVER (A,B) WITH -A=B=2 10 = -IS.00000 SADDLEPOINT •= - 0 . 1.994526 E 01 F(X)  = . 0 . 2 5 6 6 4 7 2 7 2 E-Q^  EDGEWORTH' 0 . 199 6193 530-04 0 . 1996190530-04 -0 .46629-59280-05 -0.4662.959280-05 0. 1949974560-05  CRAMER 0.4806654010-05  S A DDL F P O I M F(.X) .EDGJE^QRTH. 0.8545 185520-01 Q.854518552D-01 0 .36655.17800-01 0.366551780D-01 0 .86716.23800-01  SADDLEPOINT 2  0.2607815870-05 0.2553486560-05 0 .2567846470-05  0.2397507550-05 0.2 5683 00 81D-05 0.2558957110-05 0.2563236030-05 0.256475102D-05  +0R- 0.125F-15  0.867211958 E-01  0 .8.396239630-01  EDGEWORTH  :  SADDLEPOINT I  -5.00000 •0 .3899486 E 00 CRAMER  SADDLEPO.I NT F(X)  +OR- O.OOOE 00-  SADDLEPOINT 1  SADDLEPOINT' 2  0.114245835D 00 0.56 238 62940-01 0. 1473192.1 70 0 0  0.849,3396350-01 0.8726124220-01 0.8672 57777D-01 0.8673667630-01 0.8672415950-01  2.00000 0. 1 509085',E 00  +0R: 0.486E-15  0.705481321 E 00 .CRAMER  0 . 708 1901090 00 0.708058765D 00 0.371477.193D 00 0/. 245078542D 01 0 . 708058765D 00 0.705 5197 790 00 ,-6.1827993540 02 0 .70 5 5.1 9 7 79D_00. 6.7054803980 00  X  SADDLEPOINT F<X ) EDGEWORTH 0.99691505 ID 00 0 .9 969150 510 00 0 .9.97.49 31720 _00_ 0.907493 172 0 00 0.9975270390 00  • 10.00000 0.8983779-<E ' 00  0 .997.6713580 00  0.707837602D 00 0 .70471.38670 00 0.7054930290 00 ...0 ..7 05.46 3,2.8 9.0 00 0.7054792630 00  +0R- O.OOOE 00  -0.'..997-5308 27-E-OO — - — CRAMER  SADDLEPOINT 2  SADDLEPOINT 1  SADDLEPOINT 1 [0.997318,0520 00 0.9975994960 00 0. 997494130D 00  SADDLEPOINT 2 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 = 40  .M  X •= - 5 0 . 0 0 0 0 0 SADDLEPOINT •= - 0 . 1294131 ;E F(X) = 0 . 1 0 7 5 4 2 5 7 3  EDGEWORTH  E - 1 2  CRAMER  0 .3 78 30 3496D-•11 0 . 3 7 8 3 0 8 4 9 6 0 - • 1.1 -0 .61.4 1.9 2.4 500•11 -Q .614 1 9 2 4 5 0 0 -•11 • 0 . 4 4 5 5 3 6 3 1 9 0 - •11  +0R- O.OOOE CO  01  !  0 .2 4 2 6 7 2 3 7 7 0 - 1 2  SADDLEPOINT 1  SADDLEPOINT  0 . 1 0 8 9 0 6 9 9 2D-;]. 2 0. 1 0 7 4 5 9 7 4 4 D - 1 2 0.107 5 4 9 6 3 8 0 - 1 2  0. 10 594 1 8670 - 1 2 0 . 1 0 7 9 3 6 4 0 8 0- 1 2 0 . 1 0 7 5 4 7 ^ 4 9 0- 1 2 0. 1 0 7 5 4 3 4 ? 2 D - 1 2 0. 1 0 7 5 4 6 0 5 8 D - 1 2  2  •= - 1 0 . 0 0 0 0 0 SADDLEPOINT •= - 0 . 1892 840.E 00 . +0R- 0 . 3 4 7 E -15  x F  (  X )  ...F.QGFWD.RTH  =  .......C.R A MER  - ••  0 . 8 5 4 5 1 8 5 5 2 0 - •01 0 .854 51 3552D-•01 0 . 8 5 7 5 2 6 8 5 9 0 - •01 0 . 8 5 7 5 2 6 8 5 9 0 - •01 .01 __p.>8 5 7 56.5 C.5 3.0X  5.00000 0.9397053  0.114 1 2 7 4 3 4 0 00 0.5397668240-01 0 . 1 4 9 7 6 6 9 9 3 D 00  0 . 7 5 2 5 3 1 5 8 2  E  0 .•1519990 3 ID 01 - 0 . 4 7 5 3 3 0 9 4 3 D 01  ;  = . .0.  EDGEWORTH  00  997.O6I749..E  0 . 8 5 3 2 6 9 2 4 0 D -01 0.8589663950- -01 0.8575684]3D -01 0.857.5724120 -01 -0..857.56.5350D.-01  0 0  "SADDLEPOINTI  20.GOG00 0.3899486 .E  SADDI. EPO'I NT 2  +0R- 0 . 8 4 7 E - 15  E-01  o. 7 5.3 2 8 6 3 6 8 0 0 0 '• 0 . 5 3 8 6 3 7 3 0 9 0 00  X = SADDLE PO INT = FJ.X)  .._ SADDLEPOINT I .  CRAMER  - .  0 .7532185970 00 0 . 7 5 3 2 T 3 5 9 7 D 00 0 . 7 5 2 5 3 4 9 4 0 D 00 0....7.5 2 5? 4 9 AO D00__ . 0 . 7 5 2 5 3 1572D" 00  E - 0 1  0. 850771.582D-01  SADDLEPOINT '= F (X ) = EDGEWORTH  0 . 8 5 7 5 6 4 8 3 8  SADDLEPOINT 2 0. 7 5 3 16861 CD 0.752321958D 0 .7525322330 .0..752530474D 0 .7525315410  00 00  00 .00 00  +0R- 0 . 1 2 5 E - 1 5  .00  SADDLEPOINT  CRAMER  1  SADDLEPOINT  2  0 . 9 9 6 9 l-50-'5i?D--. 00 ••••••• 0-. 997.1245,1 ID0 0 . 0 . 9 9 6 7 5 2 3 0 0 D 00 . 0 . 9 9 7 0 8 5 9 1 1 0 0 . 9 969]50'51'D''00 ,y:- ' 0 .99.7 16 4 9 8 4 0 60 - 0 . 9 9 7 0 5 3 9 5 3 0 0 .997059581.0 00. . i . , ...l-.P.^..?.7P0.3.1 8 50 0.0 _....0.. 9 97.06 164 70 0 .9 9705 9581.0 ]00 '"""•"'.ii • i:L •.'>.'"•'•: }• \£••'• :'• 0.9 97 0 6 1 7 0 3 D o:.99706.16930: 00 .0.99706 1 7 4 5 0 ;  q  :  *•'•••';''•' •  •  00 00 00 00 00  58. Comparison w i t h the exact values f o r F ( ) x  n  tes t h a t the. s a d d l e p o i n t results  "\'3.7  indica-  2 s e r i e s again y i e l d s the most accurate  (see T a b l e s V (a) and ( b ) ) .  Remarks >. In a d d i t i o n a l t e s t cases (which f o r the sake of  •.brevity are omitted) i n v o l v i n g random v a r i a b l e s from :  . 3 . 2 to. 3 . 6 , t h e " r e s u l t s as those reported. 2.2  sections  obtained were q u a l i t a t i v e l y the same  Although f o r the reasons, c i t e d i n s e c t i o n  these methods of approximation cannot be t h e o r e t i c a l l y  justified  i n the l a t t i c e  case, d i s c r e t e random v a r i a b l e s  d i s t r i b u t e d a c c o r d i n g t o the P o i s s o n p r o b a b i l i t y law were treated.  Predictably,  the r e s u l t s were e r r a t i c and u s u a l l y  inaccurate,  but when the argument  t i n u i t y of  F  n  , the s a d d l e p o i n t  x  was a p o i n t  of d i s c o n -  2 series yielded results '  which were accurate t o two s i g n i f i c a n t f i g u r e s i n almost a l l cases. . 0  Only the Edgeworth expansion, when  , y i e l d e d r e s u l t s of s i m i l a r q u a l i t y .  c  was.close t o  59-  CHAPTER 4  OTHER APPLICATIONS.  4.1  The Non-Central Chi-Square D i s t r i b u t i o n . The  form of the c h a r a c t e r i s t i c f u n c t i o n of the non-  c e n t r a l chi-square  p r o b a b i l i t y law suggests t h a t an a l t e r n a -  t i v e approximation t o the d i s t r i b u t i o n f u n c t i o n of the fold the  convolution integrand  of t h i s lav; may be obtained  i n (1.4..4) i n powers of  the n o n - c e n t r a l i t y parameter.  \  n-  by expanding , where  \  is  The o b j e c t i v e of t h i s a l t e r n a -  t i v e approach would be an approximation which v/as u s e f u l f o r very l a r g e  \  and moderate  Let  X-^...,X  n  n  be independent, n o n - c e n t r a l c h i -  square d i s t r i b u t e d random v a r i a b l e s , each of whose t i o n has n o n - c e n t r a l i t y parameter ting;.:function of  . -  V  where  M  (  t  X.,  )  X  .  The moment genera-  ( i = 1,2,...) i s  = exp[Xt/(l-2t)] ( l - 2 t ) "  v = number of degrees of freedom and  parameter.  Equation  (1.4.4)  v / 2  !  X = non-centrality  becomes  • 1 - P ( X x ) = | ( l - s i g n ( c ) ) +~f" n  distribu-  [l-2(c+iu)]"  V n / 2  60. x exp[-Xx(c+iu) + nX(c+iu)/(l-2(c+iu)"•') ]_du_ c+iu The s a d d l e p o i n t  Thus,  c  (4.1.1).  i s the r o o t ' o f the equation  c = $(1 - Vx)  .  I f we w r i t e  g(z) = nz/(l-2z)  , and proceed f o r m a l l y  as i n Chapter 1, the i n t e g r a l i n e q u a t i o n (4.1.1) becomes  exp[-Xxc + x g ( c ) ] f  [l-2(c+iu)]"  exp[-x(-g  n v / 2  ( 2 )  (c)u /2 2  — 08 CD  ( l » )  g< >(c) r  +  E  / r J ) ] ^ H  r  r=3  T h i s can be expanded as a s e r i e s i n powers of  X  , the asy-  m p t o t i c i t y of which can be demonstrated i n a p r o o f very to t h a t of Theorem .2.1.2.  similar  T h i s expansion, up t o the f i r s t  !  five.terms, i s •  . . w : . ' ' -2Xxc F ( > x ) = | ( 1 + s i g n ( c ) ) -/-!=t 2  h  (  Q  n  0  0  p  )  + X'' (h Q (o) + h b Q ( p ) ) + X " ( h Q ( p ) ' i  1  1  1  Q  3  3  2  2  + ( h b ^ + h b ) Q ( ) + *h b|Q (p)) + X"' ?(h Q (p) . 5/  x  Q  + (h b 2  .  h  l 4 b  'T5 G 3 9 '  +  h  +  + ;  + 3  h  b  o 6 b  (i l°l h  Q  ) Q  (  6( +  4  +  4  h  b 0  5  0  0  )%(p)  x  2  h  (* 2 3  p ) +  h  b  Q  +  h  i ho lHtelo(*\ b  p  h  b  +  b  h  o 3 4 b  ( 3 3 + 2 4  +  i 3 4 b  3  ( * i 3 •+  +  ^ " ( 4 4( ).  j ) )  6  h  b  h  * o 4  +  h  b  +  b  h  b Q  b  ) Q  +  h  3 5  ^T o 3 12(p))^ h  b  Q  b  7^  5  p )  i 5 b  ) Q  8(  >  p )  (4-1.2)  61. where  . Q ( p ) ( i = 0,1, . . . , 1 2 )  = cVXg^(c)  0  i  are d e f i n e d  by equations ( 1 . 4 . 2 1 ) , ' ( 1 . 4 . 2 2 ) , (1.4.23) and (1.4.24), g b  i  h.  (c)  x  rm  = 1  ( 1 )  i  i.'Vg^(c)  * (  n  x  )  s  ^  ( i= 3,...,6)  ( i = 0,1,2,... )  i.'Vg^^c)  ,  M  denoting'the  1  M ft) = c  (l-2t)  - n v / 2  . For the sake of s i m p l i c i t y , the case considered.  ,. and .  1  0  function  =  i  y = 1  was  Numerical r e s u l t s given below i n Table VI i n d i -  cate t h a t f o r moderate v a l u e s of  n  and  \  , the expansions  are n e a r l y e q u i v a l e n t i n accuracy and speed of "convergence". As expected, our e a r l i e r approximation i s . s u p e r i o r where n and  i s large.  But even i n "extreme cases, such as when  n = 1  X = 1000, the improvement achieved by u s i n g 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  X  100  100  1000  EQUATION ( 1 . 4 . 2 5 )  n  x  15  15  8  98  F _( \x) p  EQUATION ( 4 . 1 . 2 )  .423374879  .5000000000  .428272116  .427895482  .428277904  .427895482  .428278337  .428280040  .428278338  .428280040  .138175666  .141809322  .. 1 4 5 4 4 8 5 6 1  .145367769  .145528761  .145535852  .145545801  .145545588  .145546418  .145546215  .369745696  . 373832713  .375295334  .375299472  .375308061  ,375308827  .375308876  .375308896  .375308880  .375308896  4.2  The Doubly Non-Central Let  2  x  and  F-Distribution.  2  '  2  ^  o e  • independent  w o  non-central  chi-square raiidom v a r i a b l e s w i t h degrees of freedom fg  and n o n - c e n t r a l i t y parameters  The  d i s t r i b u t i o n of  Xp = ^ i ^ i  X^ ^  s  c a  and H  e c  Xg  f^  and  , respectively.  * the doubly non-  ~2 X  central  P-distribution.  2  /  f 2  I t occurs i n the a n a l y s i s of v a r i -  ance and i s used i n e n g i n e e r i n g  problems where i t gives the  p r o b a b i l i t y of e r r o r i n c e r t a i n communications systems.  No  simple formula f o r e v a l u a t i n g  F  of  Xp  i s available.  expansions f o r F  Tiku  the p r o b a b i l i t y i n t e g r a l ,  [ 1 5 ] developed s e v e r a l  which y i e l d  ,  series  s a t i s f a c t o r y approximations  when (i) (ii) (iii) • - . .  Xj  i s l a r g e and  >,  i s small, and  1  both  X-,  and  |  and  \g  Xg  i s small  ,  i s large  ,  are l a r g e .  In t h i s s e c t i o n we o b t a i n an a l t e r n a t i v e t o T i k u ' s  approximation f o r the case when X-  \g  Xg  saddlepoint  are moderate.  f^  and  I t i s derived  method and i s , i n p a r t ,  fg  are l a r g e and .  by means of the  intended t o 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 v a r i a b l e s v/ith c h a r a c t e r i s t i c f u n c t i o n s and  cog  > r e s p e c t i v e l y , then the r a t i o  distribution function  G(x)  G  which  -l  has a  satisfies  + G(x-O) '= 1 - — r - l i m l i m T-» e-»0 n  X-,/Xg  -e T (f + \ V, ( t W - t x ) d 1 -T *V 1 ' ~~i d  (4.2.1).  co,(t) = e x p [ x , i t / ( i - 2 i t ) ] ( l - 2 i t )  Putting  o b t a i n from equation  2P(f x/f ) a  J  (j =  1,2)  J  L>  we  /2  -f  1  1  - f  n  _-  -  K  ( 4 . 2 . 1 ) , since  {J"% / }  lim lim  T  log(l-2it) - f  2  _^  I f we. suppose 'ixed, the a p p r o p r i a t e  F  f^  »  e  0  J  ^  l  i s continuous,  e x p r  VL-  V "  TT2xT  T T 2 i t  log(l+2xit)]~  while  saddlepoint  X-^Xo  equation  (4.2.  a n d  -  -g  a r e  is  0  T h i s equation  has no f i n i t e  lem occurs when we  solution,  c  .  A s i m i l a r prob -2'^1  t r y to expand i n terms of  °  r  I f the s a d d l e p o i n t method, i s to be u s e f u l , there f o r e , we  must suppose t h a t  and  are f i x e d parameters.  Xg  i y - i  f ^ -* <=  , f g -+ *>  Then  c  c =  2x  T.  f o r m a l l y at l e a s t , we  p  ~  f  ^  .  Now,  -  that  ,  < c <  ; thus,  can proceed as i n s e c t i o n (1.4)  expand the f u n c t i o n s i n the integrand  of (4.2.2) about  i n convergent power s e r i e s . Let  f ( z ) = \ z/(l-2z) - X xz/(l+2xz) p  X~^  i s ' t h e s o l u t i o n of  log(.l-2z) + - | l o g ( l + 2 x z ) ] = 0  which y i e l d s  and  , •  and c  *5. g(z) = - tf± *  Then, equation  log(l-2z) - | f  r ( )' u*  ^  2  (4.2.2)  e  1  i{c)+  *> x  p  v  [  1  ( \r  en  a  J exp[  • (r) / \ . r SLIM ( ! { ) ] . a  ro j  f( '^(c) J  exp[ s  c)  — 7 — -  -  (  i  u  )  by  j  a  j=0  a  J  and l e t bj  Proceeding (4.2.3)  = g ^(c)/j< (  ,  (j = 3 , 4 , . . . )  , and  p = c  (1.4.25),  as In the d e r i v a t i o n of equation  equation  becomes,  P ( x f / f ) = *(l+sign(c)) ~ p  1  X  ./2ir 1  f e  (  c  )  +  g  (  c  )  T. J=0  .  J  c j f ^ f ^ ) ~  (4.2.4).  Each  c .(f ,f,_,) ( j = 0 , 1 , . . . )  (f^^)  •  (4.2.4)  are  The f i r s t  J  ]  (4.2.3)  g—JTTTTJ  a. ( j = 0 , 1 , 2 , . . . )  constants  r=l  ,1  f  ^ j  x exp[ r — r M l i u ) r=3  e  f  ffn\  ? ( x f / f ) = i(l+sign(c))--  Define  ( r 2>  x  can be'"rewritten' 1  2  log(l+2xz)  2  r e p r e s e n t s a term of the order  few terms i n the s e r i e s of equation  66. c  o  (frf  2  )  = QQ(p)  c1(f1,f2)  ='fl  c2(f1,f2)  =f|  Qx(p) + ^ | Q ( P ) 5  i ' h Q2(p)+^(b4+a1b5)Q4(p)+^_^  2  c (f 5  1 S  f ) =f3 2  Q3(o) +  •^(b5+a1b4+a2b5)Q5(p)+i,f-T(b5b4  a  •  a  .  a  ci|(f1,f2)  = *4  Q4(p) +  ^(b6+a1b5+a2b4+a3b-5)Q6(p) a  a  ^ 5(|b2+b;5b +a1b3b +ia2b|)Q8(p) R  Here,  Q6(o)  Q^(o-)  5  4  ( i = 0,1,...)  +  - l ^ ^ b ^ .  i s defined as i n section 1.4.  Numerical r e s u l t s l i s t e d i n Table VII indicate that good accuracy i s achieved when X^ < 1 and  X  \^ and X  2  are small, say  . The expansion i s not, of course, uniform i n \^ 2  , and i s u s a l l y accurate to no more than 2 s i g n i f i c a n t  figures f o r larger . X^ When  f = f  •/ 2  = f  s i m p l i f i e d considerably.  , the above approximation can be  The saddlepoint  c. becomes  -^(1 - -i) .  If we l e t g ( z ) = X z / ( l - 2 z ) - X x z / ( l + 2 x z ) and h(z) = -\ l o g ( l - 2 z ) 1  - \ log(l+2xz)  2  , then  F(x) = -|(l+sign(c)) -  1  e ( g  c  )  +  f  h  (  c  )  T. c. ( f " * )  J  (4.2.5).  ' APPROXIMATIONS  TG'F-flATIO  Nl -NG.PF HTGRFES Of- FREEDOM N? =N0.OF DEGREES OF FREEDOM M . t . P . M l - N Q N CFNTRALITY PARAMFTFR N.-C.P. (2 )=MQN CENTRAL! TV PARAMETER Nl  NZ  35  40  N.C.P.(l) N.C.P.I?) 0.2500  ' 0.3000  67.  Table V I I I'M NUMERATOR N OF MOM I NA TOR IN NUMERATOR TM DENOMINATOR  T  X  F-RATIO  0.8000  0.2494549 Q 2511567 0.2501 »07 0.2 5 0 0 9 0 7 0.2 504148 u  .  20  15  0.2500  . 0.3000  0.7500  0 .27641.90 0.2 7 7 9 6 59 0/2753362 0.2725183 0.2745 522 0.5000000 • 0,494 96 5 0 0.4949650 0 .495.1 469 . 0.4951469  •  45  '40  0.7000  0.5000  1.0000  45  55  1.0000  1.2000  1.0000  25  20  I.0000  1.2000  0.8000 '  40  30  25  20  0. 1 0 0 0  0.2000  0.2 0 0 0  0.1500  • ' ' ,'•.,:'.. "/ v'v ' . •  '  :  v  •- 70 65. . .'. . . '"'  -  '  '  '."" :  0.7000'  0.8000  ••  •  • 0.1500 . • •:• 0'. lOO'O 1 .5000 • '• •••'•'• ' '-'-. "• . '' V • •• • . . • -. ' • •" • •'• • . ;  .  0.5000023 0.5011154 0. 501 1153 0.50203.13 •0.502031.3 0.3059350 0.31 820-83 0.30 7 793 0 0.. 2987 5 78 0.3041053 0.1479470 0..! 48 20 0 7 0 . 1478174 0,1474 1.33 0..1477020 0.2981952 0.2966938 0. 2 °5 87 1 9 0.2954838 0.2960377 0.9502987 0.950122.6 ' 0.9501 31 7 0.950147? 0.9501286  i  •68-.  T h e - f i r s t few c o e f f i c i e n t s of t h i s expansion are  o  c  '  =  .  c  1  = a Q ( ") + b Q ( p )  c  2  = a Q ( p ) '•+ (b^+a-jb^Q^p) + ^ ^ ( p )  1  1  2  5  P  5  2  2 c  s  = a Q ( p ) + (b +a b +a b )Q (p) + (fca-jb^+b^)  :  5  3  5  5  1  1|  2  5  5  x G^(p) + ^b|<^(p) c  4 = a Q (p) + (b +a b +a b +a b )Q (p) + 4  4  6  1  5  2  J+  5  5  6  (b^+a-jbjb^)  + t a b | + ^ ) Q Q ( P ) •+ .(fcb|b^.+ ^ a - j b ^ Q - ^ p ) + ^ b ^ O ^ p) 2  Here,  p =  J ( )( )  >  2  fh  c  section 1.4,  b  ±  =  ( i = 0,1,. .. ) i s defined i n  h ^ f c ) iWh^(c)  ( i = 3,4,... )  , and  :  a^ ( i =0,1,...)' i s defined by the equation co  (r) / \ ,  iv  ex [ i g y ^ ( r r ^ - )  ].  P  r=l  r  *  -Vh^ ' ( c )  ca  1  v  7,  s  (iy)<  a  '/ ,,j=0  J  .  Numerical calculations l i s t e d i n Table VIII again indicate - that good accuracy ^is obtained when small.  For small  f  X^  and  X g are  , approximation 4,2.5 i s considerably  more accurate than expansion 4.2.4 f o r unequal same order of magnitude. .  :  In the case when  f  i  of the  .  \^ = X g = X  approximation i s achieved f o r larger.  X  , a more suitable • say  X > 3  , if :  Table VIII . W APPROXIMATIONS TO F - R AT In ' Nl = N O . O F DEGREES OF FREFDOM IN NUMERATOR N2 = N O . O F 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 C E N T R A L ITY PARAMETER IN DENOMINATOR Nl N2 N „C . P . ( 11 N . C . P . ( 2 ) X F-RATTO .-.^  :.-^,;.; -..r. ,  ;  6a.  .  7  7  0.2500  C.0000  35  35  5.0000  1.0000  .  ,  ' * 7  7 •  0.7875  0.3708766 .0....3.6.0.L6.5.2, 0.3627365 0.3634248 0 . 3634509  0.7500 .„.._  0.1490082 Q .1.26J945Z 0 . 1199288 0 . 1236284 0 . 1245618  . 0.0000  _  0.0000  2.0000  1.0000  2.5000  0.8154488 0... .8.1.5 4 4.8.8. 0.80961.26 0.8096126 0.8096125  i  10  10  ......  L  0.0000.  _  ... ;  _  0.9357405  „  . . .  0.9376194 0.9374207 0.9373013  • 10  10  1.0000 „_„„:  1.0000' :  "  . • •' .' V  0.7500 "• • • "• •  • 9  9  • 15  15  0.5000 , w •< ' • 1.8000  0.2500 ' . *L'  0.6000 —:  . '  •• ' - ' . 1.2000 _*_  ,1.4000  '  40  40  0.2.500  ' " 6.2500  .0 . 93.8AU\3.  '  0.7875  . • '  0.3253837 .0 ..33 500.03.. 0 . 3 2 3 8 592 0.3271418 '0.3281402  0.2176499 " • 0 . ? 163377' 0.2172729 . \ " 0.2173970 0.2176049 .0.7281970 _0L. 7.0A8.SL5 7„ 0.7139909 0 . 7 1 8 52 32 0.7170775 0.3223013 __.Q...4.2 2 10.6.9 0.3909000 0.3663807 0.3724224  :  __.  70.  we  expand the i n t e g r a l i n powers of•  \ *  . ,,'  'The a p p r o p r i a t e s a d d l e p o i n t equation now  '  '  r  d  z  - |—  the i n t e g r a n d may  xz  n  < c < -|  be expanded about  remaining parameters, f  f^  f ^ < 15  , say  '  , • and  Again, the approximation  of  _  c = (-2L/x + x + l ) / ( (2,/x)(x-l))'  T h i s e q u a t i o n has r o o t I t i s seen t h a t  •  and  :  thus, the f u n c t i o n s i n c  i s not uniform  fg  i n the  , hut f o r moderate values  > f a i r l y accurate r e s u l t s  s i g n i f i c a n t f i g u r e s ) seem t o be obtained. are t a b u l a t e d i n Table  is  ( 3 or more  These and  others  IX.  4 . 3 - Remarks. The  r e s u l t s of t h i s chapter  suggest  t h a t the  saddle-  p o i n t method can be e f f e c t i v e l y a p p l i e d t o an i n t e g r a l of the ' form  [ .•  h^^(z)  g(z) e*" ( ^dz h  z  , p r o v i d e d t h a t the  -i.e.  = 0  has a f i n i t e , r e a l s o l u t i o n  ..•.v»'- '  and the i n t e g r a l can be put i n the, form where  and  . h^  o f '• X~'^  I f a s  ••'.the problem has expansion  c  f  J  00  .  I f i t does,  , g^(u)  ^ i( ) du h  e  u  s a t i s f y the c o n d i t i o n s of Theorem 2 . 1 . 1 ,  .' t h i s method w i l l - y i e l d  :  :  equation  an asymptotic  expansion  i n powers  i n the example considered i n s e c t i o n 4 . 2 ,  other parameters i n a d d i t i o n t o  X  need not be uniform w i t h r e s p e c t t o them.  , the  ,  71.  APPENDIX  Al.  •' Computing  N(z) .  .  I n t h i s appendix i s p r e s e n t e d a method o f e v a l u a t i n g N(z)  f o r complex v a l u e s o f  and Z i d e k [ 1 3 ] . N  z  w h i c h was d e v i s e d by R u b i n  I t uses c o n t i n u e d f r a c t i o n a l expansions f o r  and t h e r e b y a v o i d s t h e u n c o n t r o l l a b l e r o u n d - o f f e r r o r s  w h i c h accrue i n u s i n g t h e T a y l o r ' s e x p a n s i o n .  T h e i r method  i n v o l v e s t h e complex f o r m o f Shenton's [ 1 4 ] c o n t i n u e d f o r small values of  | z | and L a p l a c e • s c o n t i n u e d  fraction  fraction  (see K e n d a l l and S t u a r t [ 1 1 ] , p. 138) o t h e r w i s e . Since  N ( z ) = 1 - N(-z)  g e n e r a l i t y assume  a  b  Re(z)_> 0  .  , we can w i t h o u t l o s s o f  Writing  l l "2 +  :  '  a  b  2  + a^  =  • f i _ 2 • 3 . ... b,+ b + !>,+• / . > a  1  a  d  0  b^ + ...  (Al. 1),  we o b t a i n , u s i n g t h e Shenton f r a c t i o n ,  N( ):;= l; n(z) 2  :  +  vf-f.^!  r £  £  ...)  ,  „,(,).>  0  (A1.2)  and,  using Laplace's  N(z)  =1  fraction,  -,.n(z)(|-  h  h  h  ,  Re(z)  > 0 (A1.3).  Rewriting equation  (Al.2), with  t  = z  -2  1  ,  (A1.4).  We s h a l l  |_  ^  ,:„  the  2n  The  remainder  t h  call  (gn-l)/[(4n-?)(to-l)1  approximant  to  satisfies  The  l/8n ~TT  the f u n c t i o n  u(t)  u  that  is,  =  (2n+LV[ (4n+l)  on t h e  t+  (  n  right  which s a t i s f i e s  a  n  [  1  n  =  ...)  1 ) 2 )  continued f r a c t i o n  l/8n ~TT" • • •,  continued f r a c t i o n  (  (  A  1  .  5  )  in(A1.4).  '  2n/[(4n-l)(4n+l)] * 1l/8n —  the  •  ~ an(t  (4n+3) ]  \ •)  , ^ \ (A1.6).  side of the  + u)"1]-1  (A1.6) r e p r e s e n t s  equation  (A1.7),  73. u(t) = ( a  where  n  - t/2) + [ ( t / 2 )  2  (A1.8),  +  = 1/8 n Let  R (t) n  = Re{(t/2)  +  2  a } 2  (A1.9)  I ( t ) = Im((t/2) + a } 2  2  n  R ' ( t ) = t * ( R ( t ) + CR^(t) + . I ^ . ( t ) ] * ) ] * ( n = l i 2 , . . n  n  Then (see A h l f o r s  [ 2 ] , p.3)  ±  [ R ^ t ) + | I ( t ) / R ; ( t ) ] (R^(t)+0) n  '  The  square r o o t  and t h e a f u n c t i o n  i n (Al.lo)  0  otherwise  has branch p o i n t s  (A1.10) -. ;  at. + 2 a i  ,  n  o b t a i n e d by choosing e i t h e r s i g n i s a branch  of* the square r o o t .  Rather than f i x the s i g n , we take  sign[Re(tj ][R;(t) + | I ( t ) / R ^ ( t ) ] /  n  ,  Re(t)*0,R£(t)=}=0 t * l ( t ) + I„.(.t)]* = k^t) 0  +|  I (t)/R;(t) n  '  , Re(t)=0,R^(t)+0 R^(t) = 0 ( A l . 11)  to obtain  a continuous approximation t o the continued f r a c t i o n .  74. Using (A1.8) we o b t a i n , as an asymptotic approximat i o n t o the continued f r a c t i o n of (A1.4),  z_ 1-  1  2_  (2n-l)  3 t + 5- " ,  , Re(z) > 0  (4n-l)(t/2+a W(:t/2) +a ) 2  n  2  (n = 1,2,. . . )  (A1.12).  T h i s a l s o g i v e s s a t i s f a c t o r y r e s u l t s when  Re(z) = 0  J.  S i m i l a r l y , we o b t a i n an approximation t o the cont i n u e d f r a c t i o n o f (A1.3),  fe  ^  fe z+ (z +4n)  ':  R  e  (  z  )  >  °  '  <»-a.5--'> (A1.13).  One a d d i t i o n a l m o d i f i c a t i o n i s i n t r o d u c e d t o f u r t h e r improve these approximations.  H  n^  z )  =  J  ~(n-l)i  If  exp[-|(t+z) ]dt 2  , Re(z)-> 0  , (n =1,2,...) , (A1.14)  and H (z)  - e-  0  2  /  2  then H (z) n  = (n+1) H  n + 2  (z) + z H  n + 1  ( z ) - , (n = 0,1,...) (A1.15).  Letting  = H _ /H n  1  n  yields  75. Q j z ) = z + n/Q  (z)  n + 1  (Al. 16).  , (n = 1,2*.'./)  Hence,--  Q-^z) >  n ( . z ) / ( l - N ( z ) ) = z + 1/Q (z) 2  = z + 1 z+  2 z+  We- now r e p l a c e {a^}  , (n = 2,2,...)  ^(zWz  +a^)  to  a; =  where  P  _(n-ll  4n  by  2  sr (^+i)/r (|) 2  2  where the  so t h a t the approximation z = 0  exact a t  s  2 , 3 , . . . ) (A1.17),  =  i n (Al.13),  a^  , are chosen  Q^ ) ' ^  ^ ^  , that i s ,  , ( n - 2 , 5 , . . . )  ,  ( A I . I S )  denotes the Gamma f u n c t i o n . 0  Let  R ( )  and  z  n  I ( z ) denote  2, / Im(z +a^) . , r e s p e c t i v e l y .  R  and  Then, i f  (z) = [l(R (z)(-l) k  k ) n  Re(z +a^)  n  + (R^t)  1  n  l*(t))*)]  +  (k = l,.2;n = 2,3,. .. )  ,  (Al.19)  we take, as we d i d i n the d e r i v a t i o n of ( A l . l l ) , 'sign[l (z)][l (z)/2R ^ (z) + i R n  n  2  n  (z)l  2 j n  R ( z ) < 0, I (z)40,R ^ (z)+0 n  DR (z) + i l ( z ) ] * = f n ( ) / n' z  n  2 R  n  2,n( > z  n  +  1  R  2,n( > z  l,n( ) Z  >'  +  1  V >/ Z  n  2 R  2  l,n( >  n  , 2,n^ ^ R  z  o  r  R R  n  >  Z  R ( z ) > 0, S>  n  R (z)<0,I .(z) = 0, R ^ (z)+0 n  R  2  l j n  (z) + 0  l,n^ ^ z  =  0  , !  76.  Summarizing these r e s u l t s we o b t a i n  §--... (2n-l)/[ (4 n - l ) ( t / 2  N(z) ~ % + n ( z ) (jz  +  J(t/2)  2  + l/64n ):].) 2  , Re(z) >_ 0  +  l/8n (A1.21)  ,  and •  N(z) ~ i-nCzJ^^-fp' • •' ( - )/[z+732+8r (^i)/r (-§) ]) 2  n  1  2  Re(z) > 0  ,  2  ,  (Al.22)  where the square r o o t s are c a l c u l a t e d a c c o r d i n g t o ( A l . l l )  and (A1.20). The approximations were computed i n [13] for  z  comprised  of 231  p o i n t s spread over the r e g i o n  D = { z : - 5 < Re(z) _< 5,0,_< Im(z)  < _ 5}  , and Table X was com-  p l e t e d on the b a s i s of the r e s u l t s .  It lists  mations f o r d i f f e r e n t subregions of  D -.  approximations i n C n  r  (r =  1,2, ..) 0  (A1.21)  on a g r i d  and  (A1.22)  suitable approxi-  For s i m p l i c i t y the  are denoted by  , r e s p e c t i v e l y , where  r  S  i s a value of  s u f f i c i e n t l y l a r g e t o give an accuracy of at l e a s t 10  n i f i c a n t f i g u r e s over t h a t subregion.  and  r  sig-  TABLE X APPROXIMATIONS TO N(z),, z € D WHICH ARE ACCURATE TO AT LEAST TEN SIGNIFICANT PLACES REGIONS . APPROXIMATION R( ) . . KB ): ( , 3.5, . 5 ] • 5 ] V £ °> 10 ( - 4 . 2 5 , - 3 . 7 5 ] -v ;•••/.}. ( 4 . 2 5 , • ; 5 :.' ] io Z  C  c  (-4.75,  - 4 . 2 5 ] ',  0.-5,  -4.75 ].-•••• 5  ( 2.25, (-2.25,  .•(. 2 . 2 5 , .  ]'  (  -1.75'.'] ,-2"75 ]  (-4.25,  -3.75 ]  (-4.75,  -4.25 ]  [  -4.75 ]  -5  (-2.75, (  ,'v  (-1.25,  -,25 ]  (-1.75,  -1.25 ] ;  (  .75,  (-.25/ (-1.25,..  1.75 ] -  7  5  •;•=•• . . ;  '  :  ]  ( 2.25,  5.  ]  [ . 0,  5  0,  [  0,  ( - . 2 5 , •'• .75 ] (-1.25,. (-1.75,  ."•  (-2.25,  '  c  2o  c  2o  C  .  C  ]  20  20 20.  C  C  30  .25"-.]  ;.  ]  [0,  .75  ]  [0,  .25 • ] .  y - 5 •: S s  :  3.25 ' ] • 3.25  ]  2.75  ^ ( . 2 5 / 2.25  .75,  [  2o  5  -'• s i o s  io  ]  s  ]  io  s  io  s  io  s  i5  S  l5  S  15  1.25., ]  0,  :  s  5  -  :  2.75 . ] •;•  <  2  -75,  .'•  • •( 3.25, ":; "":'•//. ( 3 . 2 5 , •  -.25 ] -1-25 ] " -i.75 ]  :  •]  l • 0, .. 1.75  (  °20 c  3.25 . ]  (1.75, j'•2.25 ]  ; 1.75 ]  ;  .]  .. 4 . 2 5  / $ "•( 1 - 7 5 ,  ( - 2 . 2 5 , ; -1.75 ]  .75,  io  5  ( .25,  (-1.75,; ' - 1 . 2 5 ]'-•'.  (  io  c  [ •, 0,' • 2.. 25,' ]  -.25 ]  . 2.25 ]  5 . ]  • • [. .0, '/•< 2 . 2 5  ]  ( 1.75,  c  ]  •'[•: 0,  - 7 5 , > 1.75 ] -75-] •  ]  5  [ :  -2.25 ] .  ( - 2 5 , ..."  o,  '..•/••. ( 3 . 7 5 ,  (-2,75/ • - 2 . 2 5 ] : . (-3-75,  5  . ( 3 . 2 5 ,  :  V  ,  •  ] 5  5  :• ] •  5  ']•  :  ( 2.75,  S  ( 2.25,' (1.25,  3.75  ]  '  '  15  S  S  15  15  '•/:'; '  A2  78. .  Computer Program f o r Evaluating; the Saddlepoint 2 Approximation. To evaluate the-..saddlepoint 2 approximation  (1.4.25),  i.a computer program was written i n FORTRAN and run on the IBM 360/67 computer using the Waterloo University compiler (WATFOR). The" entire program was written i n double p r e c i s i o n to keep roundo f f - e r r o r to a minimum. Included i n t h i s appendix i s a l i s t i n g of a sample run to calculate the approximation non-central random variables.  ( 1 . 4 . 2 5 ) i n the case of  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 f o r d i f f e r e n t d i s t r i b u t i o n s of the variables X^.  In addition, the few l i n e s i n the main program (lines  47 and 48) which f i n d the saddlepoint are altered with d i f ferent cases. A sample set of data cards w i l l contain the f o l lowing information: i) ii)  •'; ./•  Cards 1 , 2 and 3 - t i t l e or comments. Card. 4 - the constant PARA, which may be any parameter that the user wishes 1  ...  the problem.  to vary during  I f PARA _> 1 0 0 0 , the program  terminates. iii)  Card 5 - an indicator showing whether the cumulants are read i n (for example, i n 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, r e s p e c t i v e l y . (iv)  Card 6 - the constant NCUM, which s p e c i f i e s the number o f cumulants t o be read i n or generated.  (v)  Cards 6 + 1, 6 + 2,... - ( o p t i o n a l ) i f Card 5 reads 2, the cummulants w i l l have t o be read i n as data,  (vi)  Card 7 - the number  ; i f n J> 1000  n  , the  program r e t u r n s t o ( i i ) . (vii)  Card 8 - the number  x  ;  i f x >_ 1000  the program r e t u r n s t o ( v i ) ; otherwise, a d d i t i o n a l v a l u e s of  x  are read i n .  }  ... i _2_ 3 4  ;  ... 80. . . SCOMPILE DIMENSION ZID(5),Q(13)„G<5),U(6) COMMON CUM<lO).STDEV.-pARAtNCUM : DOUBLE PRECISION CUM,MU1,STDEV,ENN,PI 1,PI 2,XBAR,U,PARA,X,RT,STAR DOUBLE PRECISION Q,G,W,ABSRHO,SPERR,C,DEXP,OSORT,KAY,K,KP,KPP,FI DOUBLE PRECISION.CUMUL,CEE,ESS,ARGUM,ENU,8R,RH0 ,A2,RUTN,ZI0,C0,C1 DOUBLE P R E C I S I O N TEMP EQUIVALENCE(U(1),XBAR) INTEGER OPT PI1=.3989422804014327 PI2=2.D0*PI1 -..TOL=5 . E - 1 4 „ ;..„.„.::_„ ;.., ' _ .... _ * READ IN T I T L E '.  ..  5 6 7 8 9 10 ._1L~_ C  c c 12 13 _14 15  READ(5,500) READ(5,501) . REA0(5,502).. . FORMATC70H 1 FORMAT(70H 1 FORMAT!70H ... 1 , .;,;,  500  16  501  17  502  V  ^ ......  ..,  i  1  . .. .... . ) )  \.:_.,;..„.l,_ ;,'4_,. .lx^A---~~-~ v  )  --, -~  C  c c 18 19 . 20 21  22  IN VARIABLES  •'.••^J'/V\''  22 READ15,400)PARA ; SL*-;vf A: IF{PARA.GT.999.:)G0T0. 20 :; . ' READ(5,100)0 P ^:lz^::;:^iili:v-.:...; ^....l:;..:;!^:;: .. 100 FORMAT!I 3 ) fK.'. C I F OPT={1,2).CUMULANTS ARE*(GENERATED,READ I N ) C READ(5,100)NCUM , r ;  t  -U  23 24 25 ,26 27 28 29  READ  C r  ....  MAXIMUM  u  - .......  NCUM = 7  I F I O P T . E O . n G O TO 1 R E A 0 ( 5 , 3 0 0 ) ( C U M U ),1=1,NCUM) FORMAT 13026.16) ; 300 GO TO 222 _ '• DO 3 1=1,NCUM 1 3 CUM f I ) = CUMUL(I) / 77? RFADC5.1071N  . —.  .........  ...  81. WRITE t6,499) 499 F0RMAT( 1H1) ' WRITE(6,500) WRITE (6,501) " ' ' WRITE16,502) WRITE(6,900)N 900 FORMAT<//34X,3HN =,I3//> 102 FORMAT(15)  30 . 31 32 33 34 35 36 37  ,  . ' •;  r  '0~.  • C  .. .. V.  38 39  c •r  40 41  ......... MAXIMUM  v.  START OF  c  C  \* 57 58 59 . 60 61 62 63 64 65 66 67 68 69 70 71 02  73  c c  ;  X = .999.  INITIAL  .„ ......  .  222  CALCULATIONS  '  FIND SADDLEPOINT  f  r  22  MUl=CUM(i) STDEV=DSQRT.CUM(2)) ENN=N W=(X-ENN*MUl)/(DSQRTlENN)*STDEV) XBAR=X/ENN  42 43 44 45 46  47 48 49 . 50 _ 51 52 53 54 55 56 .  TO  I F ( X . G T . 9 9 9 . ) G 0 TO FORMAT<F10„5)  400 C  c  9999  IF(N.GT.9999)G0 21 READf5,400)X  c _C. c  MAXIMUM N =  RT=o2 5D0/XBAR**2+PARA/XBAR C=.5D0*(1.-.5D0/XBAR-DSQRTIRT)) SPERR=0. 73 DO 12 1 = 2,6. ... ' 12 U ( I ) = 0. CALL UUIU,C) STAR=DSQRT(U(2)) ARGUM=-C*X+ENN*K1C) KAY=DEXP(ARGUM) *PIl ENU=ENN*U(2) . PROCEEO WITH SECOND SADDLEPOINT  ...  , . ..  APPROXIMATION  23 RHO=C*DSQRT(ENU) . / A2=RH0**2*.5D0 _,RO=RHO .__ _._ .„ :.. . ABSRHO=ABS(R 0) IFtABSRHO-2.25)38,38,37 37 Q(1)=CEE(30,ABSRH0)*SIGNUM(C> GO TO 39 38 Q ( l ) = ( D E X P ( A 2 ) / P I 2 - E S S ( 2 0 , A 8 S R H O ) ) * S I G N U M ( C ) 39 FACT=I. .. DO 10 1=1,11,2 FACT=-1.*FL0AT(I-2)*FACT Q(1+1)=FACT-RHO*QlI) 10 Q(I+2)=-RH0*Q(I+l) G(1) = Q U ) G(2)=U(3)*Q<4)/(STAR**3*6.D0) . . G ( 3 ) = U ( 4 ) * Q ( 5 ) / t U ( 2 ) * * 2 * 2 4 . D 0 ) + U ( 3 ) * * 2 * Q < 7 ) / < U ( 2)**3*72.D0) 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. 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) RUTN=l./DSQRT(ENN) . . ZIDU> = .5D0*<1.+SIGNUM(C>)-KAY*G<1> PJL_U 1 = 2 v 5 : . '. ZID(IJ=ZID(I-1)-KAY*G(I)*RUTN**{I-1)  74 75 76 TL 78  11 C  .  C . PREPARE OUTPUT C  . 1  _  .:•  _ .. „ ...  XD=X CS=C SPE=SPERR WRITE{6,1000)XD,CS,SPE 1000 FORMAT{10X,1HX,11X,IH=,F10.5/10X,13HSADDLEP0INT. =,E16.8,2X,4H+0R-, 1E10.3/10X,4HF(.X),8X,1H= /) WRITE(6,1100)(ZlOCIltl'lf5) 1100 FORMAT(4X,13HSADDLEPOINT 2/5(1X,D16.9/) ) GO TO 21 CONTINUE 20 STOP , • : END •/ . '  79 80 81 82 83 84 85 86 87 . 88 89  0  (  90 FUNCTION SIGNUMIT) ... • ' .. ' I- .DOUBLE PRECISION T . 91 .... ;.•.,.„„ 92. i..:ji,'.l.,..'..,.-..,.'J J,..-. I F I T ) 1,2,3 SIGNUM=-1. 93 ' ' 94 GO TO 4 2 SIGNUM=0. 95 GO TO 4 96 97 3 SIGNUM=1. ' '. \: : 98 .4 .RETURN....:. 99 END r  1  '  :  f  ...........  '  1  ..  :  100  DOUBLE P R E C I S I O N FUNCTION  f  NORMAL. D I S T R I B U T I O N  C ..  ....... .  . ....  _  FKT)  FUNCTION  .  . . .  C  101 102 103 104 105. . 106 107 108 109 110 111 •_. 112 113 U4 115 116 117 118 119 120  DOUBLE P R E C I S I O N DEXP,DSQRT,ESS»CEE,A,FACTOR.FF,P11,T,TT PI1=.7978845608028654 A=-.5D0*T**2 / FACTOR=.5D0*DEXPtA)*PI1 ,  .........  7 6 8 1 2 3 4 5 9  ,.TT=T  .  TSP=TT A=T IF(TT)6.7,8 FF=.5D0 GO TO 15 A=-T :,. .... IF{ABS{TSP)-1.75)1,1,2 FF=.5D0+FACT0R*ESS18,A) GO TO 5 IF<ABSlTSP)-2.2 5 ) 3 , 3 , 4 FF=.5D0+FACT0R*ESS(13,A) GO TO 5 FF=1« -FACT0R*CEE(25,A) IF(TT)9,15,15 FI=1„-FF J  .'  .  .  .  ... ...  . , .  GO TO 16 15 FI = FF 16 RETURN .., END  121  122 123 124 125  126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158  159 160 161 162 163 164 165 L66  DOUBLE PRECISION, FUNCTION ESStN,Z) C C C  SHENTON CONTINUED.FRACTION  DOUBLE PRECISION Z,DENOM,BR,RUT,T,EN,DEXP,DSQRT ESS=0. . . IF(Z.EQ.O.)RETURN EN=N : T=l./Z**2 RUT=.25D0*T**2+1./(64.D0*EN**2) : MULT=4*N-1 NUM=2*N-1 . . SIGN=-l. •• DENOM=FLOAT(MULT)*{.5D0*T+.125/EN+DSQRT{RUT)) LI M=NUM J.'..'. • ' ... . DO 1 1=1,LIM MULT=MULT-2 DENOM=FLOAT(MULT)*< (SIGN»1. ) »T-S IGN»1. ) *. 5D0-t-SI GN*FLOATt NUM) /DEN NUM=NUM-1 SIGN=-SIGN ESS=Z/DENOM . _ . RETURN END :  1  C C  DOUBLE PRECISION FUNCTION CEEJN,Z> LAPLACE CONTINUED FRACTION DOUBLE PRECISION DEXP,OSQRT,Z,DENOM,RUT,AI,A2,GAMM Al={ FLOAT{N3 + 1. ?/2.D0 A2=FLOAT<N)/2„D0 RUT=Z**2+8.*{GAMM(A1)/GAMM(A2))**2 DENOM=Z+DSQRT(RUT) . . LIM=2*N-2 DO 1 1=1,LIM : NUM=2*N-1-I ; ; DENOM=Z+FLOAT(NUM)/DENOM CEE=1./DENOM RETURN ... * . END :  1 .„  C C C  DOUBLE PRECISION FUNCTION GAMM(X) GAMMA FUNCTION DOUBLE PRECISION X,XX,FACT IF ( X.LE. 1. )G0 TO 10 N=X XX=N IF(XX.NE.X)GO TO 2 FACT=1. N1=N-1 DO t 1 = 1.Nl  :  84 167 1 . 168 169 170 2 171 17? 173 3 174 175 — „. 176 10 177 100 1 78 11 179  GAMM=FACT GO TO 11 LIM=2*N-1 FACT=1. DO 3 J=1,LIM.2 F FACT=FACT*FLOAT(J) GAMM=FACT*l.77245385O?O5516D0/2.D0**N GO TO 11 ._ _ .. .,.„„ WRITE16,100) F0RMAT(5X,20H ERROR IN GAMMA FCN. ) RETURN END  180  SUBROUTINE  r c  207 208 .... 209 210 ?ll  U,C)  •  .  AT C  DIMENSION U ( 6 ) COMMON CUM110),STDEV,PARA,NCUM DOUBLE PRECISION CUM,STDEV,U,RT,PARA,CAL,DSQRT,C RT=.25D0/U(1)**2+PARA/U(I) CAL=o5D0/U(l)+DSQRT(RT) U(2)=2./CAL**2*(i.*2.*PARA/CAL) U(3)=8.D0/CAL**3*ll.*3.D0*PARA/CAL) U(4)=48.D0/CAL**4*(1.+4.D0*PARA/CAL) U(55=384.D0/CAL**5*(l.+5.D0*PARA/CAL) U(6) = 3840.D0/CAL**6'M l.+6.*PARA/CAL) RETURN END -  DOUBLE PRECISION FUNCTION  c C C  194 195 196 197 198 199 200 201 _ 202 203 204 205  206  UU( -  DERIVATIVES OF C.G.F. CALCULATED  c 181 182 . 183 184 185 186 187 188 189 190 191 192 193  FACT=FACT*FLOAT (I ) . .  CUMUL(J)  "  CALCULATION OF CUMULANTS COMMON CUM(10),STDEV,PARA,NCUM DOUBLE PRECISION CUM,STDEV,PARA,UMUL_ UMUL=1.+ PARA: JMl=J-l FACT=1. I F ( J-1) 1, 1, 2 / 2 DO 3 K=1,JM1 ......3. FACT=FACT*2.*FLOAT{ K) .. _ *. _ UMUL=FACT*{1.+ PARA*FLOAT(J)) I CUMUL=UMUL RETURN END  J  ;  c C C  DOUBLE PRECISION FUNCTION CUMULANT GENERATING  _  _  ;  K IS-)  FUNCTION  (C.G.F.)  j  . COMMON C U M ( 1 0 ) » S T D E V , P A R A , N C U M DOUBLE PRECISION CUM,STDEV,S,PARA,ARG.DLOG . IF(S-.5)1,2,2 2 WRITE(6,100) ) 100 F0RMAT(5X.13H K UNDEFINED  '  K=0. GO TO 5 1 ARG=1„-2«.*S K=-.5D0*DL0GlARG)*PARA*S/ARG 5 RETURN END  212 213 214 215 216 217  218  DOUBLE P R E C I S I O N FUNCTION KP ( S )  r  . c c  219 220 221 222 223 224 225 226 227 228 229  ....  F I R S T DERIVATIVE  OF C.G.F.  •  COMMON CUM(10),STDEV,.PARA,NCUM DOUBLE PRECISION CUM,STDEV,S,PARA, ARG IFIS-.5)1,2,2 2 WRITE(6,100) 100 F 0 R M A T ( 5 X , 1 3 H K UNDEFINED ) K=0. GO TO 5 ARG=1.-2.*S KP=lo/ARG+PARA/ARG**2 . RETURN . ' . ." END ';: VV (  ;  " I' .  230  r  c  DOUBLE  P R E C I S I O N FUNCTION K P P ( S )  SECOND DERIVATIVE  OF C.G.F.  r  .v.  231 232 233 234 . 235 236 237 238 239 240 241  COMMON CUM(10),STDEV,PARA,NCUM DOUBLE PRECISION CUM,STDEV,S,PARA, ARG IFIS-.5)1,2,2 2 WRITEJ6,100) .. . • 100 F0RMAT(5X,13H K.UNDEFINED ) K=0. GO TO 5 1 ARG=1.-2.*S KPP=2./ARG**2+PARA*4./ARG**3 5 RETURN . ; _ _ END  /  '  ! '  86.  ' REFERENCES  [1]  Abramowitz, M. and Stegun, I.A., Handbook of Mathematical F u n c t i o n s , Dover P u b l i c a t i o n s ,  [2]  (1965).  A h l f o r s , L.V., Complex A n a l y s i s , McGraw-Hill Book Company, Inc.,  [3]  Inc., New York  New York  (1953).  Cramer, H., Mathematical Methods of S t a t i s t i c s ,  Princeton  U n i v e r s i t y Press (1945). [4]  Cramer, H.', Sur un nouveau theoreme-limite de l a t h e o r i e des p r o b a b i l i t e s : A c t u a l i t e s S c i e n t i f i q u e s e t I n d u s t r i e l l e s , No. 7 3 6 , Hermann e t C i e , P a r i s  [5]  Daniels,  H.E.,  (1938).  S a d d l e p o i n t Approximations i n S t a t i s t i c s ,  Ann. Math. S t a t . , v o l . 2 5 , p p . 6 3 1 - 6 5 0 [6]  De B r u i j n , N. G., Asymptotic Methods i n A n a l y s i s , Holland P u b l i s h i n g  [7]  (1954).  F e l l e r , W.,  (1961).  Co., Amsterdam  An I n t r o d u c t i o n  North-  t o P r o b a b i l i t y Theory and i t s  A p p l i c a t i o n s , V o l . I I , John Wiley and.Sons Inc., New York ( 1 9 6 6 ) . [8]  Gnedenko, B.V.  7  and Kolmogorov, A.N.,  Limit  Distributions  f o r Sums of Independent Random V a r i a b l e s , Addison-Wesley Publishing [9]  Co. , Cambridge, Mass.  Gurland, J . , I n v e r s i o n ratios,- Ann.,.-Math.  (1954).  formulae f o r the d i s t r i b u t i o n of  Stat., v o l . 19,  pp. 2 2 8 - 2 3 7 ( 1 9 4 8 ) .  [ 1 0 ] ' J e f f r e y s , H. and J e f f r e y s , B.S., Methods i n Mathematical Physics,  Cambridge U n i v e r s i t y P r e s s  (1950).  87.  [11]  K e n d a l l , M. G. and S t u a r t , A., The Advanced Theory of S t a t i s t i c s , V o l . I, C. G r i f f i n and Co., London  ['12]  (1958).  Parzen, E., Modern P r o b a b i l i t y Theory and i t s A p p l i c a t i o n s , John Wiley and Sons, Inc., New York ( i 9 6 0 ) .  •[13]  Rubin, H. and Zidek, J. , Approximations t o the D i s t r i bution F u n c t i o n of Sums of Independent C h i Randoirr V a r i a b l e s , T e c h n i c a l 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  [14]  Shenton, L.R., I n e q u a l i t i e s f o r the normal  (1965). integral  i n c l u d i n g a new continued f r a c t i o n , B i o m e t r i k a , V o l . 4 l , pp. 177-189  [15]  (1954).  T i k u , M.L. S e r i e s expansions f o r the doubly n o n - c e n t r a l F - d i s t r i b u t i o n , A u s t r a l i a n Journal of S t a t i s t i c s , V o l . 7,  [16]  pp. 7 8 - 8 9 ,  Sydney ( 1 9 6 5 ) .  Watson, G. N., Theory o f B e s s e l F u n c t i o n , Cambridge v e r s i t y Press ( 1 9 4 8 ) .  Uni-  

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