Stochastic Orders Comparisons of Negative Binomial Distribution with Negative Binomial Lindley Distribution

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1 Oen Jounal of Statcs 8- htt://dxdoog/46/os5 Publshed Onlne Al (htt://wwwscrpog/ounal/os) Stochac Odes Comasons of Negatve Bnomal Dbuton wth Negatve Bnomal Lndley Dbuton Chooat Pudommaat Wna Bodhsuwan Deatment of Statcs Kasetsat Unvey Bango Thaland Emal: Receved Januay ; evsed Febuay 8 ; acceted Mach 4 ABSTRACT The uose of ths udy s to comae a negatve bnomal dbuton wth a negatve bnomal Lndley by usng ochac odes We chaacteze the comasons n usual ochac ode lelhood ato ode convex ode exectaton ode and unfomly moe vaable ode based on theoem and some numecal examle of comasons between negatve bnomal andom vaable and negatve bnomal Lndley andom vaable Keywods: Stochac Odes; Negatve Bnomal Dbuton; Negatve Bnomal Lndley Dbuton Intoducton The negatve bnomal (NB) dbuton s a mxtue of Posson dbuton by mxng the Posson dbuton and gamma dbuton The NB dbuton s emloyed as a functonal fom that elaxes the ovedseson (vaance s geate than the mean) ecton of the Posson dbuton (see []) If X denote a andom vaable of NB dbuted wth aamete and then ts obablty mass functon s n fom x f x x x x fo and wth EX and Va X The negatve bnomal Lndley (NB-L) dbuton whch s a mxed negatve bnomal dbuton obtaned by mxng the negatve bnomal dbuton wth a Lndley dbuton The NB-L dbuton was ntoduced by Zaman and Ismal n [] and t ovdes a model fo count data of nsuance clams If Y s a NB-L andom vaable wth aamete and then ts obablty mass functon s n fom y y y gy y y fo and wth EY when and Va Y 4 when In ths esect the am of ths wo s to comae a negatve bnomal dbuton wth negatve bnomal Lndley dbuton base on ochac odes such as usual ochac ode lelhood ato ode convex ode exectaton ode and unfomly moe vaable ode Stochac Odes Stochac odes ae useful n comang andom vaables measung cetan chaactecs n many aeas Such aeas nclude nsuance oeatons eseach queung theoy sval analyss and elablty theoy (see []) The smle comason s though comang the exected value of the two comaable andom vaables The followng we wll defne some notons of the ochac odes whch wll be used n the context of the ae Fo moe detals we efe to Ross [4] Msa [5] Shaed [67] and Sngh [8] Defnton Let X and Y be andom vaables wth denes f and g esectvely such that g f s non-deceasng functon n ove the unon of the suots of X and Y o eqalently f ugv fvgu fo all u v Then X s smalle than Y n the lelhood ato ode whch s denoted by X Y l Coyght ScRes

2 C PUDPROMMARAT ET AL 9 Defnton Let X and Y be two andom vaables such that PY PX fo all Then X s smalle than Y n the usual ochac ode whch s denoted by X Y Defnton Let X and Y be two andom vaables such that EX EY fo evey eal valued convex functon whee exectatons ae assumed to be exed Then X s smalle than Y n convex ode whch s denoted by Xcx Y Defnton 4 Let X and Y be two andom vaables such that EX EY whee exectatons ae assumed to be exed Then X s smalle than Y n the exectaton ode whch s denoted by XE Y Defnton 5 Let X and Y be two andom vaables wth denes f and g esectvely Recall that su(x) and su(y) denote the esectve suot of X and esectve suot of Y such that su(x) su(y) and the ato f g s a unmodal functon ove su(y) Then X s smalle than Y n unfomly moe vaable ode whch s denoted by X Y Comason We mae comasons between the negatve bnomal andom vaable and negatve bnomal Lndley andom vaable wth esect to the lelhood ato ode ochac ode convex ode exectaton ode and unfom moe vaable ode The followng lemma wll be useful n ovng the man esults Lemma Defne a ( ) m m m and m m Then ) a s a non- nceasng functon of ) Fo each fxed m s concave functon of m Poof ) We may wte fo a that whee a h e e h ; e e h ; d d E W s the Lndley dbuton defned by h z; z z e z and and W s a andom vaable havng the obablty deny functon: z z e e hz; z ( ) z Fo fxed the ato x x s obvously a non-nceasng functon of x Then by Defntons and we have W l W whch yelds W W and theefoe EW EW o eqalently a a Ths oves a s a non-nceasng functon of ) Fo note that m m s both convex and concave Fo we can wte m m m m m () The elatonsh between negatve bnomal and beta obabltes s of the fom! t t!! Theefoe m Thus m n Equaton () can be wtten as m m m! m t t!!! m m m!! m m whch oves concavty Theoem Let X~NB Y~NB-L and Th en Futhemoe ) X l Y f an d only f ) X Y f and only f ) X E EY f and only f Poof ) The lelhood ato ode between Y and X can be wtten as Coyght ScRes

3 C PUDPROMMARAT ET AL P Y l P X By Defnton we have Xl Y ll+ + a a Snce a s non-nceasng n (by at ) n Le mma ) then whch ovdes a necessay and suffcent cono n fo the l n Equaton () to be non-deceasng Ths comletes the oof of the esult ) Let X Y b y Defnton we have P Y P X Convesely suose that Fo consde PX PY X~ NB and X~NB and f Xl X Hence we get Consequently X X then and theefoe Fo fxed ex ~ Lndley We get and () E P E P E P P Usng concave functon (by at ) n Lemma ) can be wtten as EP E P Alyng Jensen s nequalty to concave functon we have E P E P Convesely whee X Y m- Ths oves ) The oofs of the esults ae obvous Theoem Suose that fo evey t P t then ) No value of ca n ensue that X Y ) X Y f an d only f Poof P X ) We fnd R by e- P Y and les that P Y P X fo fyng the numeato and denomnato as foll owng: P X!!!!!!!!! t t t Fo any P Y e e h ; d e e h ; d e! t t!! h ; d log t! t t h ; d!!! t t Hlog ;!! t Coyght ScRes

4 C PUDPROMMARAT ET AL Table Stochac odes comasons of NB andom vaables wth NB-L andom vaables Random Vaables Ode Comasons of NB Random Vaables wth NB-L Random Vaables NB X~NB 8 NB-L Usual ochac ode Lelhood ato ode Convex ode Exectaton ode Unfomly moe vaable ode Y~N B-L5 X Y X Y l 5 - X Y E Y ~ N X Y X Y E - - Y~NB-L X Y X Y E Y ~ NB-L X Y X Y E 4 4 B-L cx 5 Y ~ NB-L 57 5 X Y X Y X Y E 5 5 whee H s cumulatve dbuton functon of Lndley dbuton: z z Hz; e z and! P Y H ; t!! So! H ;!! R H; Snce lm we have H ; then lm R Theefoe t follows that fo any thee exs a suffcently lage such that PX PY Ths valdates the esult ) Suose that The n fom at ) n Theoem and at ) n Theoem t s clea that andom vaables X and Y ae not odeed by the usual ochac ode Also fom the aguments used n the oof of a t ) n Theoem snce t follows that P X P Y s non-nceasng and unmodal mlyng that X Y The convese at follows by usng the smla aguments Theoem Suose that Then ) X Y ) X cx Y Poof ) Follows fom at ) n Theoem we have X Y whee ) Snce X Y and EY E X by the esult of Shaed n [4] X Y cx Next We shows some numecal examles of the comasons between negatve bnomal andom vaable wth negatve bnomal Lndley andom vaable n usual ochac ode lelhood ato ode convex ode exectaton ode and unfomly moe vaable ode and the esults ae ovded n Table Then we exlan that negatve bnomal andom vaable (X) s smalle than negatve bnomal Lndley anode mles dom vaable (Y) n the usual ochac that X E Y In adon f X and Y have esectve suots su(x) and su(y) such that su(x) su(y) and the ato P X P Y s a unmodal functon ove su (Y) but X and Y ae not odeed n t he usual ocha c ode Futhemoe f X and Y have a same mean Then X Y mles that X Y cx 4 Concluson Ths ae shows ocha c odes comason of negatve bnomal andom vaable wth a negatve bnomal Lndley andom vaable by usual ochac ode lelhood ato ode convex ode exectaton ode and unfomly moe vaable ode Some advantages of ochac odes comason between negatve bnomal andom vaable and negatve bnomal Lndley andom vaable ae as follows: If negatve bnomal andom vaable (X) s smalle than negatve bnomal Lndley andom vaab le (Y) n the usual ochac ode Its us efulness s that t gves a smle suffcent conon fo X s smalle than Y n the exectaton ode Next f su(x) su(y) s that t mles that the ato P X P Y s a unmodal functon ove su(y) but X and Y ae not odeed n the usual ochac ode Fnally If X and Y have a same mean t s nown that X s smalle than Y n unfomly moe vaable ode m- convex ode Ths con- les that X s smalle than Y n cluson s suoted by numecal examles 5 Acnowledgements We ae gateful to the Commsson on Hghe Educaton Mny of Educaton Thaland fo fundng suot unde the Stategc Scholashs Fellowshs Fonte Reseach Netwo Coyght ScRes

5 C PUDPROMMARAT ET AL REFERENCES [] W Rane Econometc Analyss of Count Data d Eon Snge-Velag Beln [] H Zaman and N Ismal Negatve Bnomal Lndley Dbuton and Its Alcaton Jounal of Mathematcs and Statcs Vol 6 No 4-9 do:844/mss49 [] M Shaed and J G Shanthuma Stochac Odes Academc Pess New Yo 6 [4] S M Ross Stochac Pocesses Wley New Yo 98 [5] N Msa H Sngh and E J Hane Stochac Comasons of Posson and Bnomal Random Vaables wth The Mxtues Statcs and Pobablty Lettes Vol 65 No do:6/sl7 [6] M Shaed On Mxtues fom Exonental Famles Jounal of the Royal Statcal Socety: Sees B Vol 4 No [7] M Shaed an d J G Shanthuma Stochac Odes and The Alcatons Academc Pess New Yo 994 [8] H Sngh On Patal Odengs of Lfe Dbutons Naval Reseach Logcs Vol 6 No do:/5-675(989)6:<::aid-nav 68>CO;-7 Coyght ScRes