 Open Journal of Statistics, 2012, 2, 208-212 http://dx.doi.org/10.4236/ojs.2012.22025 Published Online April 2012 (http://www.SciRP.org/journal/ojs) Stochastic Orders Comparisons of Negative Binomial Distribution with Negative Binomial—Lindley Distribution Chookait Pudprommarat, Winai Bodhisuwan Department of Statistics, Kasetsart University, Bangkok, Thailand Email: chookait.pu@hotmail.com Received January 20, 2012; revised February 18, 2012; accepted March 4, 2012 ABSTRACT The purpose of this study is to compare a negative binomial distribution with a negative binomial—Lindley by using stochastic orders. We characterize the comparisons in usual stochastic order, likelihood ratio order, convex order, ex-pectation order and uniformly more variable order based on theorem and some numerical example of comparisons be-tween negative binomial random variable and negative binomial—Lindley random variable. Keywords: Stochastic Orders; Negative Binomial Distribution; Negative Binomial—Lindley Distribution 1. Introduction  The negative binomial (NB) distribution is a mixture of Poisson distribution by mixing the Poisson distribution and gamma distribution. The NB distribution is em-ployed as a functional form that relaxes the overdisper-sion (variance is greater than the mean) restriction of the Poisson distribution (see ). If X denote a random variable of NB distributed with parameter r and p then its probability mass function is in form rxfx  xr1p 1 px0p1, , x0,1,2, for and , with r0r1 pEXp and  2r1XppVar . The negative binomial—Lindley (NB-L) distribution which is a mixed negative binomial distribution obtained by mixing the negative binomial distribution with a Lind- ley distribution. The NB-L distribution was introduced by Zamani and Ismail in  and it provides a model for count data of insurance claims. If Y is a NB-L random variable with parameter r and  then its probability mass function is in form  2yjry1 ygy 12j0rj1yj rjr 0, y0,1,2,, for and , with  3r11 12rEY , when , and 22 23222223224rr1 2rrVar Y12 11r11 r,11     2 when . In this respect, the aim of this work is to compare a negative binomial distribution with negative binomial— Lindley distribution base on stochastic orders such as usual stochastic order, likelihood ratio order, convex or-der, expectation order and uniformly more variable order. 2. Stochastic Orders Stochastic orders are useful in comparing random vari-ables measuring certain characteristics in many areas. Such areas include insurance, operations research, queu-ing theory, survival analysis and reliability theory (see ). The simplest comparison is through comparing the expected value of the two comparable random variables. The following, we will define some notions of the sto-chastic orders which will be used in the context of the paper. For more details, we refer to Ross , Misra , Shaked [6,7] and Singh . Definition 1. Let X and Y be random variables with densities f and g, respectively, such that gkf k is non-decreasing function in k over the union of the sup-ports of X and Y, or, equivalently, fugvfvguuvlrXY, for all . Then X is smaller than Y in the likelihood ratio order which is denoted by . Copyright © 2012 SciRes. OJS C. PUDPROMMARAT ET AL. 209Definition 2. Let X and Y be two random variables such that , for all . Then X is smaller than Y in the usual stochastic order which is denoted by . PrYkXYXkPr kEYstDefinition 3. Let X and Y be two random variables such that , for every real valued convex function EX where expectations are assumed to be existed. Then X is smaller than Y in convex order which is denoted by . cxDefinition 4. Let X and Y be two random variables such that , where expectations are as-sumed to be existed. Then X is smaller than Y in the ex-pectation order which is denoted by . XY XYEEXY EDefinition 5. Let X and Y be two random variables with densities f and g, respectively. Recall that supp (X) and supp(Y) denote the respective support of X and re-spective support of Y, such that supp(X) supp(Y) and the ratio fk gkuvXY is a unimodal function over supp(Y). Then X is smaller than Y in uniformly more variable order w hi c h is denoted by . 3. Comparison We make comparisons between the negative binomial random variable and negative binomial—Lindley random variable with respect to the likelihood ratio order, sto-chastic order, convex order, expectation order and uni-form more variable order. The following lemma will be useful in proving the main results. Lemma 1. Define, k1 jj0kjj0k1 1jak 1k(1)j 22rj1rjrj1rj and j1r1mm1mk0,1,2,akk0,1,2,2,kmk0,1,2,kkj0jrj, m, , Then, 0m11) is a non- increasing function of , 2) For each fixed , is concave function of . k0,1,m0,1Proof. 1) We may write for , that ak kdEWd hk1r0kr0e1e h;ak 1e1e h;, where is the Lindley distribution defined by  2zzehz; 11 and 0, z0 and kW is a random variable havinprobaility density function: g the b krz zkk1e1eh z;z2j2j0krj1(1)jrj , z0. For fixed k0,1,2,, the ratio k1 kxx is obviously a non-increasing function of x0. Then, by 1 and 2, welrk1 Definitions have kWW, which yields kst k1WW and therefore EWkk1EW  or, equivalently, akak 1. This proves ak is a non-increasing function of k0,12) For k0,2,. , note that mm is bothex and concave. For k1,2,0 convrite , we can wj1jrkjk1jr1m1m1m, m, 0m1. (1) The relationship between negative binomial and betaprobabilities is of the form 1pjr1rk1jkkr1!rr10p1pt1t dtjk  , 1!r1!k0,1,2,. Therefore, km in Equation (1) can be written as   1r1m0m1mr1kkkr! t1tdtk!r1 !, , 0m1. Thus, k11121rrkkr!1mm1m0rr 1!r!2m, m, 0m1. which proves concavity. □ Theorem 1. Let X~NB r,p, Y~NB-L r, and 2r2 rp 03r1, 12r12r1pr, 22311p. Th 2100pp p 1Furthermore, 1) lrXYen . if an 0d only ifp, p2) stXY if and only if 1pp, 3) XEly if 2EY if and onpp, of. he likeltwn Y and X can be written as Pro1) Tihood ratio ordeeer beCopyright © 2012 SciRes. OJS C. PUDPROMMARAT ET AL. 210   2kkrkrj1,X kp1p 1k0,1,2,. (2) By Definition 1, we have  j2j01jPr Y kr jlk Pr  lrk1 jj0kjj00XYlklk+1, , 220,1,2, ,1rj11rj p1,krj11jrj pak, , 0,1,2,, ppa0.kkkk     Since k+jak is non-increasing in k (by part 1) in e, then 0Lmma 1)pp which provides a necessary and conditio for the sufficientnlk in Equation (2) to be o b nn-decreasing. This completes the proof of the result. 2) Let stXYy Definition 2, we have 2r2r1PrY 0PrX 0p,r 12r12r1 pp.r  Conversely, suppose10p p1 that . k0  For ,1,2,, considerkiri02kij2i0 j0ir1kp 1prj1ir1i 1,ij 1rj   X~1X~NB r,p, then 1st 1XX. qki111pkPrX kPrYi and if lrXXConseNB r,p and nce, we get ly, . Heuent kirri0 i0ir1 irp1p p1piikk   ,  pp . For fixed and therefore 1k0,1,2,, 12r2r1r, 1ppexp and ~ Lindley. We get   1kkpirp11i0kij2i0 j01pirj1ir1i 1,ij 1rj  1i1krrrpi0kiri0ir1kEP1EPiir1 EP1P,i  Using concave function (by part 2) in Lemma 1), 1pk can be written as 1rrpk kkEPEP  . Applying Jensen’s inequality to concave function, we have rrEPE P and k0 for 2ir1kk1pk, 0,1,2,. Conversely, where XYkst at im-plies thX 0PrY 0Pr. This proves 1pp. 3) The proofs of the results are obvious. □ Theorem 2. Suppose that, for every 0t1, tp 1Pr0, then 1) No value of 0p1 ca that XYn ensurest, 2uvXY)  if an. d only if 10pp 1 Proof. 1) We find Pr Xk0,1,Rk Pr Yk, k, btor and 2,y re- frdowing:  ying the numeraenominator as foll jrjk1pr1k1 r10rk1p ,jkr1! t1 tdt,k1!r1! 1p1tdt,k1!r1!1p 1pkr1! .k1!r1!r1pk For any 10pp 1k1 01pkr1! jr1pPr X, k0,1,2,,  2ij2ik j0ir0irik01e r1k10rj1ri1iPr Yk1,ij rjri1 e1eh;d,iri1 e1eh;d,ikr1! t1tdtk1!r1!    ik  01log 1t1r1k1001r1k10h; d,kr1! t1th;ddt,k1!r1!kr1! 1 t1tHlog;dt,k1!r1! 1t     Copyright © 2012 SciRes. OJS C. PUDPROMMARAT ET AL. Copyright © 2012 SciRes. OJS 211 Table 1. Stochastic orders comparisons of NB random variables with NB-L random variables. Random Variables Order Comparisons of NB Random Variables with NB-L Random Variables NBr, p NB-Lr, Usual stochastic orderLikelihood ratio orderConvex orderExpectation order Uniformly more variable orderB-L 3,1.51Y~N stlr 5XY 1EXY - 2Y~N st 2XY 2EXY - 3Y~NB-L3,3.5 - - - E3XYuv 3XY 4Y~NB-L3,4.5 - - - EXYuv 4XY 5Y~NB-L3,5.7 EXYuv 5XY 1XY - B-L3,2.0 - - 4 X~NB 3,0.8 - - cx 5XY5 where H is cumulative distribution function of Lindley distribution: z1ze1Hz; 1, z0 and 0.  11pr1 1kr111kr1!Pr YkHdt,1!Hp; p 1kt 110k1!r1! p ;prk1!r11p.k ! kSo,   kr111p1pRk 1pr1 pp Hp;. r1k1Since  kr11k1 p1plim 01pr1 ppH;, we havethen r1k1p klim Rk0, 1pp1. lidates the result. ose that The part 2) in 2, it is clear that ran-domd Y are not ordered by the usual sto-chastfrom the arguments used in the proof of pa 1, since 0Thert follows that, for any 0p1, there ex-ists a sufficiently large k such that PrXk PrYkefore, i. This va10p prt 1) in Theorem2) SuppTheorem1.n, from 1 and pa variables X anic order. Also, rt 1) in Theorempp it follows that Pr s non-increasing and unimodal, hat . The converse part follows by mints. □ hat 2XkPrYk iuvXYlar argume Suppose timplying tusing the siTheorem 3.pp. Then, uv2) cxXYows from part 2) in Theorem 2, we ha, where 11) XY . Proof. 1) Follve uvXY2pp. uvXY and 2) Since  32r1rEYr Xp11 , pEbyed in , Xwith negative binomial—Lindley random variable in usual stochastic order, likelihood ratio order, convex or-der, expectation order and uniformly more variable order and the results are provided in Table 1. Then, we explain that negative binomial random vari-able (X) is smaller than negative binomial—Lindley ran-order implies that XYdom variable (Y) in the usual stochastic E. In addition, if X and Y have respective supp(X)  supports supp(X) and supp(Y), such thatsupp(Y) and the ratio PrX kPrYk is a uni-modal funtion over supp but X and Ye not or-dered in t usual stochast order. Furthermre, if X and Y have a same mean. ThenuvXY implthat cxXYche(Y)ic aro ies . 4. Conclusion This paper shws stochast orders comparisof nega-tive binomial random variable with a negative binomial— , exd uniformly more variabcompariinomial—Lindley random vanegatiial random vale (X) is smaller than negative binomial—Lindley ram variab (Y) in the usual stochastic order. Its us is that it gives a simple sufficient condition for Xxt, if supp(X) (Y) is tt it implies that the ratio oicn oLindley random variable by usual stochastic order, like-lihood ratio order, convex orderpectation order anle order. Some advantages of sto-chastic orders son between negative binomial random variable and negative briable are as follows: If ve binomriabndo leefulness is smaller than Y in the expectation order. Ne suppha the result of Shakcx Y. Next, We shows some numerical examples of the comparisons between negative binomial random variable PrPr YkXknimodal function over supp(Y) but X and Y are not ordered in the usual stochasticr. Finally, If X and Y have a same mean, it is known X is smaller than Y in uniformly more variable order im-n converder. This con-clR is a u orde thatplies that X is smaller than Y ix ousion is supported by numerical examples. 5. Acknowledgements We are grateful to the Commission on Higher Education, Ministry of Education, Thailand, for funding support under the Strategic Scholarships Fellowships Frontier esearch Network. C. PUDPROMMARAT ET AL. 212 REFERENCES  W. Rainer, “Econometric Analysis of Count Data,” 3rd Edition, Springer-Verlag, Berlin, 2000.  H. 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