几何分布样本间隔的似然比序.doc

几何分布样本间隔的似然比序张艺赢 , 赵鹏兰州大学数学与统计学院，兰州730000摘要：文献中，对于指数分布的样本间隔已有大量的研究结果。众所周知，几何分布是指数分布的离散对应情形，但目前为止尚未见到几何分布的样本间隔的任何结果。本文在似然比序意义下研究了几何分布的样本间隔的随机比较问题，具体来说，我们建立了如下结果：当两组几何参数向量满足弱超优序时，其对应的样本间隔之间满足似然比序，此结果实际上与指数情形是相互对应的。关键词：几何分布; 样本间隔；似然比序；超优序中图分类号： O211.9Likelihood Ratio Order on Sample Ranges ofGeometric distributionZHANG Yiying , ZHAO PengSchool of Mathematics and Statistics, Lanzhou University, Lanzhou 730000Abstract: In the literature, there exists much work on sample ranges from exponentialdistributions. It is well known that the geometric distribution is the discrete counterpart ofexponential distribution. However, there is no any result on sample ranges so far. In thispaper, we investigate the likelihood ratio order of sample ranges arising from geometricdistribution. Specically, it is shown that, if the weak majorization order holds between thetwo parameter vectors, then the likelihood ratio order holds between the correspondinggeometric sample ranges. This result established here actually corresponds to the exponentialcase.Key words: Geometric distribution; Sample ranges; Likelihood ratio order; Majorization0IntroductionOrder statistics and those statistics having a close relation to order statistics play a promi-nent role in many areas of probability and statistics. In particular, spacings are of great interest基金项目： National Natural Science Foundation of China (11001112)，Research Fund for the Doctoral Programof Higher Education (20090211120019).作 者 简 介： ZHANG Yiying（1988-），male，Master student，major research direction：Applied Probabilityand Statistics. Correspondence author：ZHAO Peng（1980-），male，associate professor，major research direction：Applied Probability and Statistics, reliability theory, statistical inference.-2-in many areas such as goodness-of-t tests, reliability theory, auction theory, actuarial science,life testing, operations research, information sciences, and many other areas. One may re-fer to Balakrishnan and Rao (1998a, 1998b) for some goodness-of-t tests based on functionsof sample spacings. Let X1:n X2:n Xn:n denote the order statistics arising fromrandom variables X1, X2, . . . , Xn. Then, the k-th order statistic Xk:n is just the lifetime of a(n k + 1)-out-of-n system, which is a very popular structure of redundancy in fault-tolerantsystems that have been studied extensively. In particular, Xn:n and X1:n correspond to thelifetimes of parallel and series systems, respectively.Due to the nice mathematical form and the unique memoryless property, the exponentialdistribution has widely been used in many elds including reliability analysis. One may refer toBarlow and Proschan (1975) and Balakrishnan and Basu (1995) for an encyclopedic treatmentto developments on the exponential distribution. There is a large number of papers in theliterature on stochastic comparisons of exponential sample spacings; see, for example, Kocharand Xu (2011b) for a review on this topic. Recently, some researchers carried out stochasticcomparisons of sample ranges wherein one sample is homogeneous while another sample isheterogenous. Let X1, . . . , Xn be independent exponential random variables with Xi havinghazard rate i, i = 1, . . . , n. Let Y1, . . . , Yn be a random sample of size n from an exponentialdistribution with common hazard rate . Then, Kochar and Rojo (1996) showed, for =ni=1 i/n, thatXn:n X1:n st Yn:n Y1:n,(1)where st denotes the usual stochastic order and the formal denitions of various stochasticorders used in this paper will be given in next section. Zhao and Li (2009) strengthened thisresult and presented the following equivalent characterization: Xn:n X1:n st Yn:n Y1:n,where(ni=1i)1/(n1).Kochar and Xu (2007) improved the result in (1) from the usual stochastic order to thereversed hazard rate order asXn:n X1:n rh Yn:n Y1:n.Genest et al. (2009) proved, for = , thatXn:n X1:n lr Yn:n Y1:nandXn:n X1:n disp Yn:n Y1:n.-3- =Mao and Hu (2010) further presented the following equivalent characterizations: Xn:n X1:n lr Yn:n Y1:n Xn:n X1:n rh Yn:n Y1:n.Recently, Zhao and Zhang (2012) and Ding et al. (2013) further obtain some new results underthe exponential framework along this line.It is well known that the geometric distribution is unique discrete distribution that alsopossesses memoryless property. However, there is no any result presented in the literature withrespect to the sample ranges under the geometric setup.Let X1, X2 be independent geometric random variables with parameters p1, p2, respectively.And Y1, Y2 be another set of independent geometric random variables with parameters p1, p2,respectively. Suppose that p1 p1 p2 p2, then, it is shown thatwIn contrast to the exponential case, the result established here provides a new attempt fordealing with the related problems of sample ranges arising from geometric distribution, whichcan be applied in the area of actuarial science and auction theory.1DenitionsIn this section, we recall some notions of stochastic orders, and majorization and relatedorders. Throughout this paper, the term increasing is used for monotone non-decreasing anddecreasing is used for monotone non-increasing.Stochastic ordersDenition 1. For two random variables X and Y with densities fX and fY , and distributionfunctions FX and FY , respectively, let F X = 1 FX and F Y = 1 FY be the correspondingsurvival functions. Then:(i) X is said to be smaller than Y in the likelihood ratio order (denoted by X lr Y ) iffY (x)/fX(x) is increasing in x;(ii) X is said to be smaller than Y in the hazard rate order (denoted by X hr Y ) ifF Y (x)/F X(x) is increasing in x;(iii) X is said to be smaller than Y in the reversed hazard rate order (denoted by X rh Y ) ifFY (x)/FX(x) is increasing in x;(iv) X is said to be smaller than Y in the stochastic order (denoted by X st Y ) if F Y (x) F X(x);-4-(log(1 p1), log(1 p2) (log(1 p1), log(1 p2) = X2:2 X1:2 lr Y2:2 Y1:2.(v) X is said to be smaller than Y in the mean residual life order (denoted by X mrl Y ) ifEXt EYt, where Xt = (X t|X t) is the residual life at age t 0 of the randomlifetime X;It is known that likelihood ratio order implies both usual stochastic order and hazard rateorder, but neither usual stochastic order nor hazard rate order order implies the other; seeShaked and Shanthikumar (2007).Majorization and related ordersThe notion of majorization is quite useful in establishing various inequalities. Let x(1) x(n) be the increasing arrangement of the components of the vector x = (x1, . . . , xn).Denition 2. (i) A vector x = (x1, . . . , xn) n is said to majorize another vector y =mjx(i) jy(i)for j = 1, . . . , n 1,i=1i=1andifn ni=1 x(i) = i=1 y(i);wjx(i) jy(i)for j = 1, . . . , n;i=1i=1p(iii) A vector x n+ is said to be p-larger than another vector y n+ (written as x y)ifjx(i) jy(i)for j = 1, . . . , n.i=1i=1m w p wObviously, x y implies x y, and x y is equivalent to log(x) log(y), where log(x)m pis the vector of logarithms of the coordinates of x. Note that x y implies x y for anypx, y n+. The converse is, however, not true. For example, (1, 5.5) (2, 3), but clearly themajorization order does not hold between these two vectors.For more details on majorization and p-larger orders and their applications, one may referto Marshall et al. (2011) and Bon and Paltanea (1999).2Likelihood Ratio OrderingLemma 3. (Marshall et al., 2011) Let I be an open interval and let : I n becontinuously dierentiable. Then, is Schur-convex Schur-concave on I n if and only if is-5-(y1, . . . , yn) n (written as x y) if(ii) A vector x n is said to weakly majorize another vector y n (written as x y)symmetric on I n and for all i = j,zizj(z) 0 for all z I n,where (z) denotes the partial derivative of (z) with respect to its i-th argument.ziTheorem 4. Let X1, X2 be independent geometric random variables with parameters p1, p2, re-spectively, and Y1, Y2 be another set of independent geometric random variables with parametersp1, p2, respectively. Then, if p1 p1 p2 p2 andmwe haveX2:2 X1:2 lr Y2:2 Y1:2.Proof: To obtain X2:2 X1:2 lr Y2:2 Y1:2, we need to prove thatfR(X)(k)fR(Y )(k)fR(X)(k + 1)fR(Y )(k + 1), k = 0, 1, 2, . . . ,i.e.,fR(X)(k)fR(X)(k + 1)fR(Y )(k)fR(Y )(k + 1), k = 0, 1, 2, . . . .(2)Denote(p1, p2) =fR(X)(k)fR(X)(k + 1),(3)we then divide the proof into two cases.Case I: k = 0In this case, (3) becomes(p1, p2) =fR(X)(0)fR(X)(1)=12 p1 p2.Let 1 = log(1 p1), 2 = log(1 p2), it suces to show that the symmetric function : (0, )2 (0, ) given by(1, 2) =e11+ e2is Schur-concave. In fact, it is easy to check that(1 2) (1, 2)1 (1, 2)2 0,which means that (1, 2) is Schur-concave, i.e.,fR(X)(0)fR(Y )(0)fR(X)(1)fR(Y )(1).(4)Case II: k 1-6-(zi zj)(z) (log(1 p1), log(1 p2) (log(1 p1), log(1 p2),Upon using the method of Case I, we need to prove that the symmetric function :(0, )2 (0, ) given by(1, 2) =ek1 + ek2e(k+1)1 + e(k+1)2is Schur-concave. Taking the derivative of (1, 2) with respect to 1 gives rise to (1, 2)1=e(2k+1)1 + (k + 1)e(k+1)1ek2 kek1e(k+1)2e + e(k+1)2.Similarly, (1, 2)2=e(2k+1)2 + (k + 1)ek1e(k+1)2 ke(k+1)1ek2e + e(k+1)2.Then, we have (1, 2)1 (1, 2)2sgn=e(2k+1)1 e(2k+1)2 + (2k + 1)e(1+2)k (e1 e2)sgn=2 10,which implies that(1 2) (1, 2)1 (1, 2)2 0.Thus, we obtain thatfR(X)(k)fR(Y )(k)fR(X)(k + 1)fR(Y )(k + 1), k = 1, 2, . . . .To sum up, it holds thatX2:2 X1:2 lr Y2:2 Y1:2.Theorem 5. Let X1, X2 be independent geometric random variables with parameters p1, p, re-spectively. Let Y1, Y2 be another set of independent geometric random variables with parametersp2, p, respectively. Suppose p maxp1, p2. Then the necessary and sucient condition forX2:2 X1:2 lr Y2:2 Y1:2is p1 p2.Proof: We just consider the suciency, the proof of necessity is the inverse process of thesuciency. Using the similar means in Theorem 4, we complete the proof of suciency by twocases.Case I: k = 0-7- (k+1)1 2 (k+1)1 2From the fact that p1 p2 p, we havefR(X)(0)fR(X)(1)=12 p1 p12 p2 p=fR(Y )(0)fR(Y )(1).Case II: k 1According to Case II of Theorem 4, it suces to prove the function(1) =ek1 + eke(k+1)1 + e(k+1)is increasing in 1 (0, . Taking the derivative of (1) with respect to 1 gives rise to 0, which states thatfR(X)(k)fR(Y )(k)fR(X)(k + 1)fR(Y )(k + 1), k = 1, 2, . . . .Hence, the suciency follows.Corollary 6. Let X1, X2 be independent geometric random variables with parameters p1, p2,respectively. Let Y1, Y2 be another set of independent geometric random variables with parame-ters p1, p2, respectively. Then, if p1 p1 p2 p2 andwwe haveX2:2 X1:2 lr Y2:2 Y1:2.Proof: The desired result holds for the case (1 p1)(1 p2) = (1 p1)(1 p2) according toTheorem 4. We now just consider the case when (1 p1)(1 p2) (1 p1)(1 p2). Thereexists a p satisfying that p1 p p1 and (1 p)(1 p2) = (1 p1)(1 p2). Denote Z1, Z2 beanother set of independent geometric random variables with parameters p, p2. On the basis ofTheorem 4, it follows thatZ2:2 Z1:2 lr Y2:2 Y1:2.On the other hand, upon using Theorem 5, we haveX2:2 X1:2 lr Z2:2 Z1:2.Thus, it holds thatX2:2 X1:2 lr Y2:2 Y1:2.-8-(1) = kek1 e(k+1)1 + e(k+1) + (k + 1)e(k+1)1 ek1 + ek= e(2k+1)1 + eke(k+1)1 + kekek1 ek1 ek(log(1 p1), log(1 p2) (log(1 p1), log(1 p2),Corollary 7. Let X1, X2 be independent geometric random variables with parameters p1, p2,respectively. Let Y1, Y2 be another set of independent geometric random variables with a commonparameter p. Then, the necessary and sucient condition forX2:2 X1:2 lr Y2:2 Y1:2isp 1 (1 p1)(1 p2).参考文献（References）1 Balakrishnan, N. and Basu, A. P. (Eds.) (1995). The Exponential Distribution: Theory,Methods and Applications. Newark, New Jersey: Gordon and Breach.2 Balakrishnan, N. and Rao, C. R. (Eds.) (1998a). Handbook of Statistics. Vol. 16: OrderStatistics: Theory and Methods. Amsterdam: Elsevier.3 Balakrishnan, N. and Rao, C. R. (Eds.) (1998b). Handbook of Statistics. Vol. 17: OrderStatistics: Applications. Amsterdam: Elsevier.4 Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing:Probability Models. Silver Spring, Maryland: To Begin With.5 Bon, J. L. and Paltanea, E. (1999). Ordering properties of convolutions of exponentialrandom variables. Lifetime Data Analysis 5, 185-192.6 Ding, W., Da, G. and Zhao, P. (2013). On sample ranges from two sets of heterogenousrandom variables. Journal of Multivariate Analysis 116, 63-73.7 Genest, C., Kochar, S. C. and Xu, M. (2009). On the range of heterogeneous samples.Journal of Multivariate Analysis 100, 1587-1592.8 Kochar, S. C. and Rojo, J. (1996). Some new results on stochastic comparisons of spacingsfrom heterogeneous exponential distributions. Journal of Multivariate Analysis 59, 272-281.9 Kochar, S. C. and Xu, M. (2007). Stochastic comparisons of parallel systems when com-ponents have proport