New bounds on Poisson approximation to the distribution of a sum of negative binomial random variables

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1 Sogklaaka J. Sc. Tchol. 4 () 4-48 Ma Ogal tcl Nw bouds o Posso aomato to th dstbuto of a sum of gatv bomal adom vaabls * Kat Taabola Datmt of Mathmatcs Faculty of Scc Buaha Uvsty Muag Chobu 3 Thalad Ct of Ecllc Mathmatcs Commsso o Hgh Educato Ratchathw Bagkok 4 Thalad Rcvd: Novmb 6; cctd: Jauay 7 bstact Th St-Ch mthod s usdto gv w bouds o-ufom bouds fo th dstacs btw th dstbuto of a sum of ddt gatv bomal adom vaabls ad a Posso dstbuto wth ma wh ad a aamts of ach gatv bomal dstbuto. Rsults of ths study a suo tha thos std Taabola (4) ad Hug ad Gag (6). Kywods: gatv bomal dstbuto Posso aomato o-ufom boud St-Ch mthod. Itoducto I obablty thoy ad statstcs th gatv bomal dstbuto wth aamts R ad () s a motat dsct dstbuto wth a log hstoy as sam as th bomal dstbuto. Wh N t s calld th Pascal dstbuto wth aamts N ad () ad wh t s calld th gomtc dstbuto wth aamt. Not that th gatv bomal dstbuto ca b cosdd as a mtu of a Posso dstbuto wth a gamma mg dstbuto (Kals & Xkalak 5). I addto som sach tocs latd to Posso aomato otd out that th Posso dstbuto wth ma *Cosodg autho Emal addss: kat@buu.ac.th o s a good aomato of th gatv bomal dstbuto wth aamts ad wh s small whch ca b foud Vvaat (969) Romaowska (977) Gb (984) Roos (3) ad Taabola (). Howv ou tst s aomatg thdstbuto of a sum of ddt gatv bomal adom vaabls by a Posso dstbuto whch s th ma cott of ths study. Lt S X wh X... X a ddt adom vaabls followg th gatv bomal dstbutos ach wth th obablty mass fucto ( ) ( ) X ( )! N {}. Lt Z dot a Posso adom vaabl wth ma ( ). Fom th cocluso mtod abov ad fo ach {... } t follows that f s small th th gatv bomal dstbuto wth aamts ad s aomatd by a Posso dstbuto wth ma

2 K. Taabola / Sogklaaka J. Sc. Tchol. 4 () o. ddtoally w kow that th dstbuto of a sum of ddt Posso adom vaabls ach wth ma s th Posso dstbuto wth ma thus t s aoat to aomat th dstbuto of S by a Posso dstbuto wth ma ES ( ) o wh all a small. I th ast fw yas som authos havto gv ufom ad o-ufom bouds o Posso aomato to th dstbuto of S wth both Posso mas as follows. ad Fo th Posso ma Vllasamy ad Uadhy (9) usd th mthod of ots to gv a ufom boud fo (.) N {} wh d S Z P( S ) P( Z ) s th dstac btw th dstbuto of S ad a Posso dstbuto wth ma.i th cas of cumulatv obablty aomato Taabola (7b) usd th St- Ch mthod to gv ufom ad o-ufom bouds fo th ato btw th cumulatv dstbuto fucto of ad th Posso cumulatv dstbuto fucto fo ad th fom (.) ( ) PS (.3) f wh ( ) P( Z ) f (. ) ad. Fo N Hug ad Gag (6) usd th St-Ch mthod to gv two o-ufom bouds th followg foms: Fo S P( S ) m P( Z ) N {} ad P S su P( Z ) N {} P( Z ) ( ) ( )! m P S P ( Z ) ( ( )) ( ) P Z P Z K d ( S Z ) m (.4) (.5) wh d S Z P( S ) P( Z ) ad ( ). I th cas of otws aomato Taabola (67a) usd th St-Ch mthod to gv a ufom boud th fom ( ) f d S (.6) Z P( Z) P( Z) ma f fo wh Fo th Posso ma Taabola (4) usd th StCh mthod ad w-fuctos to gv a ufom boud th fom (.7) fo N {}. Fo Hug ad Gag (6) usd th St-Ch mthod to gv a ufom boud as follows: (.8) ad thy also gav a o-ufom boud fo cumulatv obablty aomato th fom fo (.9). I th cas of otws aomato Taabola (5a) usd th sam tools as Taabola (4) to gv a o-ufom boud th fom fo. K N (.) W obsv that th boud (.8) s wos tha that (.7) bcaus t caot b ald to th cas ad N v though t may b sha tha that (.7). Futhmo both bouds (.7) ad (.8) do ot chag alog N {} whch may b aoat fo masug th accuacy of th aomato.notc that th boud (.9) caot b ald th cas ad N. I ths a w am to dtm w bouds o-ufom boudswth sct to th bouds (.7)-(.9). d S Z P( S ) P( Z ) N d S Z m ( ) m ( ) K N {} m N

3 44 K. Taabola / Sogklaaka J. Sc. Tchol. 4 () Mthod I 97 St toducd a ow full mthod fo th omal aomato whch s calld St s mthod. Lat Ch (975) dvlod ad ald St s mthod to th Posso aomato whch s calld th St-Ch mthod. St s uato fo Posso dstbuto wth ma fo gv h s of th fom h( ) P ( h) f ( ) f ( ) (.) wh ad f ad h a boudd al valud fuctos dfd o. Fo N {} lt f h( ) f. b dfd by (.) Followg Babou t al. (99) th soluto f of (.) ca b ssd as P ( h) h( k) k h : N {} R ( )! [ P ( ) ( hc ) P ( h ) ( )] f P hc f f (.3) wh N ad C {... }. Smlaly fo C ad N {} s of th fom ( )! [ P( hc ) P ( )] f hc (.4) fc ( ) ( )! [ P ( h ) ( )] f C P h C f. Lt f C k k! N {} f ( ) f ( ) f ( ) ad fc ( ) fc ( ) fc ( ) fo gvg th dsd sults w also d th followg lmma. Lmma. Lt N m{ } ad ma{ C } th w hav th followg: ). Fo f ad N {} f( ) m (.5) wh s tak to b wh ad fo t s gv by ad f ( ). f f (.6) ). Fo f C ad N ( ) m C ( ) f (.7) ad f (.8) C ( ). Poof. ) Th ualty (.5) follows dctly fom Taabola (5b) ad ualty (.6) follows fom Babou t al. (99). ). Fo w hav ma{ C } ad thus (.5) bcoms ( ) m C f. (.9) Taabola (7) showd that C ( ) f ( ). (.) Combg th bouds (.9) ad (.) th boud (.7) s obtad ad fally th boud (.8) ca b obtad fom th boud (.6). 3. Ma Rsults Th ma ot of ths study s to dtm w bouds o-ufom bouds fo two dstacs d S Z ad dk S Z. Th followg thom gvs o dsd sult. C Thom 3. Lt N {} ad th w hav th followg ualty. (3.) Poof. Substtutg ad h by S ad h sctvly ad w tak ctato to (.) ylds d S Z m m.

4 E f ( S ) S f ( S ) wh K. Taabola / Sogklaaka J. Sc. Tchol. 4 () E f ( S ) X f ( S) E f ( S ) X f ( S) (3.) s dfd (.3). Fo... lt S S X th w obta f f E f ( S ) X f ( S) E f ( S X ) X f ( S X ) E E f ( S X ) X f ( S X ) X E f ( S X ) X f ( S X ) X X ( ) E ( ) () ( ) ( ) () f S X E f S f S X E ( 3) ( ) () ( 4) 3 ( 3) (3) f S f S X E f S f S X E f ( S ) E f ( S ) E f ( S ) 3 ( ) ( 3) ( ) ( ) 4 3 ( )( ) ( )( ) ( 3! 4) E f S E f ( S 3) 3 ( ) ( ) ( ) 4 ( ) 3 ( 3) ( ) ( 3) 5 4 ( )( ) ( )( ) 3! ( 4) ( 4) 3 E f ( S ) E f ( S ) E f ( S ) 4 3 ( ) 3 E f ( S ) E f ( S 3) ( ) E f ( S 3) ( )( ) ( )( ) ( ) ( 3) ( 3! 4) E f S E f S 5 ( )( ) E f ( S 4) E f ( S 4) E f ( S ) E f ( S ) 3 3 ( ) E f ( S ) ( ) E f ( S 3) 4 4 ( )( ) ( )( ) (by ) ( 3) ( 4) 5 5 ( )( )( 3) ( )( )( 3) 3! 3! ( 4) ( 5) 3 E f ( S ) f ( S ) ( ) E f ( S ) f ( S 3) 4 5 ( )( ) ( )( )( 3) ( 3) ( 4) f S 3! ( ) ( ) ( )! f S ( )! f S E f ( S 4) f ( S 5) ( ) ( ) ( ) ( 3)

5 46 K. Taabola / Sogklaaka J. Sc. Tchol. 4 () ( 3) 3 ( 4) 4 ( )3! f S ( )4! 3 ( 3) ( 4) 4 E f ( S 4) f ( S 5) X ( ) E ( ) ( ) f S f S Puttg th sult (3.3) to (3.) w hav that X Bcaus by (.5). (3.3) ( ) E f ( S ) f ( S ) X ( ) E f ( S ) f ( S ) X ( ) E f ( S ) X j. ( ) f ( j ) P( S j) X j ( ) f ( j ) P( S j) X j ( ) m P ( S j ) EX m ( ) (3.4) m (3.5) Ths gvs th Thom 3.. Fo cumulatv obablty aomato t s otd that th cas w ca comut th act o- bablty of that s P( S ). So ths cas a w o-ufom boud fo d S Z wh N s as follows. Thom 3. Lt ad th th followg ualty holds: ( ) K d S Z m m. (3.8) Poof. Usg th sam agumts dtald as th oof of Thom 3. togth wth Lmma.() th sult (3.8) s obtad. Rmak ) By comag th bouds (.7) (.8) ad (3.) t s s that S m m N m m ad K ad by (.6) X j ( ) f ( j ) P( S j) X ( ) ( ) j P S j j X ( ) ( ) P S j j X ( ) (3.6) thus fom (3.5) ad (3.6) w obta X j ( ) f ( j ) P( S j) m m. (3.7) Substtutg th boud (3.7) to (3.4) t follows that m m m ad th boud (3.) ca b ald fo all cass of whch s wd tha th boud (.8). Thfo th sult (3.) s btt tha thos std (.7) ad (.8). Smlaly th sult (3.8) s also btt tha that std (.9). ) If w comb th sults (3.) ad (.) th a w o-ufom boud fo wh N s of th fom It s a slghtly movmt of (.). (3.9) Fo aomatg th dstbuto of a gatv bomal adom vaabl X wth aamts R ad by d S Z m m.

6 () K. Taabola / Sogklaaka J. Sc. Tchol. 4 () a Posso dstbuto wth ma w ca aly th sults Thoms 3. ad 3. ad (3.9) to gv w sults as follows. Coollay 3. Fo ) Fo ) Fo ad th w hav th followg. (3.) (3.). (3.) Poof. Bcaus all sults (3.)-(3.) ca b obtad by usg smla mthod t suffcs to show th sult (3.). lyg (3.) w hav d X Z m m m. (3.3) Bcaus by Taylo s aso [( ) ] [( ) ]!! hav w ad whch mls that. Thfo th ualty (3.3) duc to d X Z m. Fom whch th sult (3.) s ovd. If th ad th sults Thoms 3. ad 3. ad (3.9) bcom to b th sults th Posso aomato fo a sum of ddt gomtc adom vaabls whch st th followg coollay. N {} Coollay 3. If ad th w hav th followg. ) Fo N {} m d X Z N m m K d X Z d X Z d S Z m m. (3.4) ) Fo ad N m m ( ) K d S Z m m. 4. Coclusos (3.5) (3.6) Th w bouds o-ufom bouds ths study w obtad by usg th St-Ch mthod. Each boud ca b usd to aomat th o of th dstac btw th dstbuto of a sum of ddt gatv bomal dstbuto ad a Posso dstbuto wth ma ES ( ) as wll wh all a small. Futhmo by comag th sults ths study ad th sults Taabola (4) ad Hug ad Gag (6) t ca b cocludd that th sults ths studyasuotha thos std Taabola (4) ad Hug ad Gag (6). Rfcs Babou. D. Holst L. & Jaso S. (99). Posso aomato (Ofod studs obablty ). Ofod Eglad: Clado Pss. Ch L. H. Y. (975). Posso aomato fo ddt tals. als of Pobablty Gb H. U. (984). Eo bouds fo th comoud Posso aomato. Isuac Mathmatcs Ecoomcs Hug T. L. & Gag L. T. (6). O bouds Posso aomato fo dstbutos of ddt gatv bomal dstbutd adom vaabls. SgPlus 5 -. Kals D. & Xkalak E. (5). Md Posso Dstbutos. Itatoal Statstcal Rvw Romaowska M. (977). ot o th u boud fo th dstac total vaato btw th bomal ad th Posso dstbutos. Statstca Nladca

7 48 K. Taabola / Sogklaaka J. Sc. Tchol. 4 () Roos B. (3). Imovmts th Posso aomato of md Posso dstbutos. Joual of Statstcal Plag ad Ifc St C. M. (97). boud fo th o omal aomato to th dstbuto of a sum of ddtadom vaabls. Pocdgs of th Sth Bkly Symosum o Mathmatcal Statstcs ad Pobablty Taabola K. (). Th last u boud o th Posso-gatv bomal latv o.commucatos Statstcs-Thoy ad Mthods Taabola K. (4). Posso aomato fo ddt gatv bomal adom vaabls.itatoal Joual of Pu ad ld Mathmatcs Taabola K. (5a). Potws Posso aomato fo ddt gatv bomal adom vaabls.global Joual of Pu ad ld Mathmatcs Taabola K. (5b). Nw o-ufom bouds o Posso aomato fo ddt Boull Tals. Bullt of th Malaysa Mathmatcal Sccs Socty Taabola K. (7a). o-ufom boud o Posso aomato fo a sum of gatv bomal adom vaabls. Sogklaaka Joual of Scc ad Tchology 39(3) Taabola K. (7b). Posso aomato fo a sum of gatv bomal adom vaabls. Bullt of th Malaysa Mathmatcal Scc Socty 4() Vllasamy P. & Uadhy N. S. (9). Comoud gatv bomal aomatos fo sums of adom vaabls. Pobablty ad Mathmatcal Statstcs Vvaat W. (969). U boud fo dstac total vaato btw th bomal o gatv bomal ad th Posso dstbuto. Statstca Nladca