An Asymptotic Expansion for the Non-Central Chi-square Distribution. By Jinan Hamzah Farhood Department of Mathematics College of Education

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1 A Asypoic Expasio fo h o-cal Chi-squa Disibuio By Jia Hazah ahood Dpa of Mahaics Collg of Educaio 6 Absac W div a asypoic xpasio fo h o-cal chi-squa disibuio as wh X i is h o-cal chi-squa vaiabl wih dg of i fdo ( ) ad Though cify fo advaag of his asypoic w foud ou ha a algoih basd o his xpasio ca b o asily ud fo paicula accuacy quis ad fo paicula paa ags 6 ( ) X i i

2 Ioducio Th o-cal chi-squa disibuio is dfid by(hy959;moodgaybill &Bos974) as h followig If X X X a idpdly disibud ad i X is ) ( i h h ado vaiabl i X i is calld a o-cal chi-squa vaiabl wih dg of fdo W call / i i h o caliy paa of h disibuio ad is h o-cal paa Th disibuio fucio is giv by \ P wh d d P is icopl gaa fucio aio d is icopl gaa fucio ad d is gaa fucio \ d () which is h cuulaiv disibuio fucio of o-cal chi-squa disibuio A o-cal chi-squa disibuio sudid by oh sachs such as(sakaa96;rub974;vabls975;sigl979;adso98;ala98;eis&joh so99) whos wokd i ou fild by akig h diff subjc of his disibuio such as :Appoxiaios o h o cal chi-squa disibuio Esiaio of h ocaliy paa of a chi-squa disibuio Maxiu liklihood siaio i h o-cal chi-disibuio calculaio of cofidc ivals fo o-cal disibuios W ioduc ay sachs who usd h asypoic xpasio i ou fild as h fo Th uifo asypoic xpssios fo saddl poi igals Applicaio o a pobabiliy disibuio occuig i ois hoy divd by (Ric968) ad h uifo

3 asypoic xpasio fo a class of polyoials biohogoal o h ui cicl divd by (T986) A icopl laplac igals: uifo asypoic xpasio wih applicaio o h icopl Ba fucio sudid by(t987) ad h uifo asypoic xpasios of Lagu polyoials divd by (z & Wog988) Th galizd icopl gaa fucios sudid by(chaudhy & Zubai994)as wll as h asypoic ad closd fo of galizd icopl gaa fucio divd by (ChaudhyT ad Vlig996) A uifo asypoic xpasios fo Mix polyoials divd by(ji & Wog998) ad h asypoic of h hypgoic fucio ioducd by (Jos) - Moivaig xapl h icopl gaa fucio Cosid h followig icopl gaa fucio : d This fucio is dfid lik h gaa fucio xcp ha h scod vaiabl appas i h upp lii of igaioobviously ad Aoh lad fucio is also calld h icopl Gaa fucio: d wh d is h odiay gaa fucio I is asy o chck ha - Divaio of a Asypoic Expasio fo h o-cal Chi-squa Disibuio Cosid h igal i () calld icopl gaa fucio To fid a asypoic xpasio of h as by xpadig h fucio i pows of ad igaig by w hav d d d

4 4 sig h aio s w s ha h sis covgs fo all Howv h sis is oly usful fo sall (fo lag valus of i covgs vy slowly)to dal wih fo lag w pocd idicly wih ad fixd Rah ha aylo xpadig h igad i (as w did abov)wi () d d W ow ak h icopl gaa fucio d W shall dvlop h igal by paig h igaio by pas: d d d wh h aid d R

5 5 R Th igads of succssiv igals a bcoig sall i gio of igaio Howv his sis gad by igaio by pas dos o covg fo fixd fii (by usig h aio s)bu fo fixd h o coid by oiig R is sall fo lag ha is as R fo fix-d I od o pov his w sia R as follows : Assu fo dfiiss h lags of i h ag of igaio is ( d d d o as Thus fo sufficily lag h aid R will b sall ad oly a fw s i h sis a dd o giv a asoabl appoxiaio o Sis sus lik ha a calld asypoic xpasios ad a wi as:

6 6 By subsiuio i quaio () w g as () alhough h sis is divg ow wh w subsiu h foula () i quaio () w g \ Th by usig h idiy w hav \ as (4) his sul givs a asypoic xpasio fo h o-cal chi-squa disibuio

7 Rfcs Ala K & Saxa L (98) Esiaio of h ocaliy paa of a chi-squa disibuio Aals of Saisics:-6 Adso D A (98) Maxiu liklihood siaio i h o-cal chi disibuio wih ukow scal paa Sakhya Sis B4b:58-67 Chaudhy M AT M & Vlig E J M (996) Asypoic ad closd fo of a galizd icopl gaa fucio Joual of Copuaioal ad Applid Mahaics 67 :7-79 Chaudhy M A & Zubai S M (994) Galizd icopl gaa fucios wih applicaios Joual of Copuaioal ad Applid Mahaics 55 :99-4 Eis D M & Johso L (99) o-cal ad cal chi-squa ad ba disibuio fucios as spcial cass of h disibuio of a idfii quadaic fo Couicaios i Saisics-Thoy ad Mhods : z C L & Wog R (988) ifo asypoic xpasios of Lagu polyoials SIAM J Mah Aal9 :-48 Hy S (959) Th aalysis of vaiac Joh Wily ad sos Ic w Yok Ji X S & Wog R (998) ifo asypoic xpasios fo Mix polyoials Cosuciv Appoxiaio 4 :-5 Jos D S () Asypoic of h hypgoic fucio Mah Mhods Appl Sci 4 :69-89 Mood A M Gaybill A & Bos D C (974) Ioducio o h hoy of saisics d d Mc Gaw-Hill Ic w Yok Ric S O (968) ifo asypoic xpasio fo saddl poi igals Applicaio o a pobabiliy disibuio occuig i ois hoy Bll Sys Tchical Joual 47 :97- Rub H (974) o-cal chi-squa ad gaa visidcouicaio i Saisics : 67-6 Sakaa M (96) Appoxiaios o h o-cal chi-squa disibuio Bioika5 :99-4 Sigl A (979) Th o-cal chi-squad disibuio wih zo dgs of fdo ad sig fo uifoiy Bioika66 :8-86 T M (987) Icopl laplac igals :uifo asypoic xpasio wih applicaio o h icopl ba fucio SIAM J Mah Aal8 :68-66 MR & 9f: 46 T M (986) ifo asypoic xpasio fo a class of polyoials biohogoal o h ui cicl Cos Appox :69-76 Vabls W (975) Calculaio of cofidc ivals fo o-cal disibuios Joual of h Royal Saisical Sociy Sis B 7 :46-4 7

F(F \m 1,m 2, ), which is suitable for large, small, r N and 0<q<1, where. is the incomplete Gamma function ratio and

$F(F \m 1,m 2, ), which is suitable for large, small, r N and 0<q<1, where. is the incomplete Gamma function ratio and$ A Asyptotic Expasio fo th No-tal -Distibtio y Jia azah ahoo Dpatt of Mathatics ollg of Ecatio 6 Abstact A asyptotic xpasio is iv fo th o-ctal -istibtio (\ ) hich is sitabl fo lag sall N a