非齐次gamma样本的最大次序统计量的随机不等式.doc

非齐次gamma样本的最大次序统计量的随机不等式赵鹏兰州大学数学与统计学院，兰州730000摘要：本文研究了基于非齐次gamma样本的最大次序统计量的随机比较问题，在反失效率意义下建立了随机不等式。所取得的结果加强和推广了文献中已有的一些结果。一些数值例子被用来阐述所取得的结果。关键词：可靠性理论; gamma分布；反失效率序；Majorization；并联系统中图分类号： O211.9A Stochastic Inequality on the Largest OrderStatistics from Heterogeneous Gamma VariablesZHAO PengSchool of Mathematics and Statistics, Lanzhou University, Lanzhou 730000Abstract: In this paper, we compare the largest orders statistics arising from independentheterogeneous gamma random variables according to the the reversed hazard rate order. Theresult derived here strengthens and generalizes some of the results known in the literature.Some numerical examples are also provided to illustrate the main result.Key words: Reliability theory; gamma distribution; Reversed hazard rate order;Majorization; Parallel system0IntroductionOrder statistics play an important role in statistical inference, reliability theory, life testing,operations research, and many other areas. Let X1:n Xn:n denote the order statistics ofrandom variables X1, . . . , Xn. The k-th order statistic Xk:n then corresponds to the lifetime ofa (n k + 1)-out-of-n system, which is a very popular structure of redundancy in fault-tolerantsystems that have been studied extensively in the literature. Xn:n and X1:n correspond tothe lifetimes of parallel and series systems, respectively, which are the simplest examples ofcoherent systems. Many papers have appeared on various aspects of order statistics when the基金项目： National Natural Science Foundation of China (11001112)，Research Fund for the Doctoral Programof Higher Education (20090211120019).作者简介： ZHAO Peng（1980-），male，associate professor，major research direction：Applied Probability andStatistics, reliability theory, statistical inference.observations are independent and identically distributed (i.i.d.). The case when observationsare non-i.i.d., however, often arises in a natural way in real life. Due to the complexity of thedistribution theory in this case, not much work is available in the literature; see, for example,David and Nagaraja (2003), Balakrishnan and Rao (1998a, 1998b), and the recent review articleof Balakrishnan (2007) for comprehensive discussions on the independent and non-identicallydistributed (i.ni.d.) case.The exponential distribution, due to its nice mathematical form and the unique memorylessproperty, has widely been applied in many elds. Pledger and Proschan (1971) were the rstto compare stochastically the order statistics arising from i.ni.d. exponential random variables.After that, many researchers have devoted themselves on this topic, including Proschan andSethuraman (1976), Kochar and Rojo (1996), Dykstra et al. (1997), Khaledi et al. (2011),Khaledi and Kochar (2000, 2007), Bon and Paltanea (2006), Kochar and Xu (2007), Paltanea(2008), Sun and Zhang (2005), Zhao and Balakrishnan (2009, 2011), and Zhao et al. (2008,2009). Gamma distribution is one of the most commonly used distributions in reliability andlife testing. Let X be a gamma random variable with shape parameter r and scale parameter. Then, in its standard form X has the probability density functionf(x; r, ) =r(r)xr1ex,x 0.It is an extremely exible family of distributions with decreasing, constant, and increasingfailure rates when 0 r 1, respectively. In this paper, we will focus onthe largest order statistic from gamma observations, i.e., the lifetime of a parallel system withindependent heterogeneous gamma components. The results established in this paper extendthe associated ones in the literature from exponential to gamma distributions.Let X1, . . . , Xn be independent exponential random variables with Xi having hazard ratei, i = 1, . . . , n. Let X1, . . . , Xn be another set of independent exponential random variableswith Xi having hazard rate i . Pledger and Proschan (1971) proved, for 1 k n, thatm(1)Proschan and Sethuraman (1976) generalized the result in (1) from componentwise stochasticordering to multivariate stochastic ordering, and Boland et al. (1994) pointed out by a coun-terexample that (1) can not be strengthened from usual stochastic order to hazard rate orderwhen n 2. Khaledi and Kochar (2000) partially improved the result in (1) asp(2)Let Y1, . . . , Yn be a random sample of size n from an exponential distribution with hazard ratei=1(1997) then showed thatXn:n hr Yn:n,(3)(1, . . . , n) (1, . . . , n) = Xk:n st Xk:n.(1, . . . , n) (1, . . . , n) = Xn:n st Xn:n.n = i/n and denote by Yn:n the corresponding largest order statistic. Dykstra et al.which was further strengthened by Kochar and Xu (2007) asXn:n lr Yn:n.(4)On the other hand, Khaledi and Kochar (2000) also strengthen the result in (3) under a weakercondition that if Z1, . . . , Zn are a random sample of size n from an exponential distribution1i=1 i) n , thenXn:n hr Zn:n.(5)This paper aims to study such a topic as above under gamma framework. We prove, ifX1, . . . , Xn are independent gamma random variables with Xi having shape parameter 0 r 3.8 and scale parameter i, i = 1, . . . , n, and X1, . . . , Xn are another set of independent gammarandom variables with Xi having shape parameter r and scale parameter i , i = 1, . . . , n, thenm(6)which strengthens the result of Sun and Zhang (2005) from usual stochastic order to reversedhazard rate order. Apparently, the result in (6) substantially strengthens and generalizes thosefor the exponential case established earlier in the literature.1DenitionsIn this section, let us 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. For two random variables X and Y withdensities fX and fY , and distribution functions FX and FY , respectively, let F X = 1 FXand F Y = 1 FY be the corresponding survival functions. X is said to be smaller than Yin the likelihood ratio order (denoted by X lr Y ) if fY (x)/fX(x) is increasing in x; X issaid to be smaller than Y in the hazard rate order (denoted by X hr Y ) if F Y (x)/F X(x) isincreasing in x; X is said to be smaller than Y in the reversed hazard rate order (denoted byX rh Y ) if FY (x)/FX(x) is increasing in x; X is said to be smaller than Y in the stochasticorder (denoted by X st Y ) if F Y (x) F X(x). It is well known that likelihood ratio orderimplies hazard rate order reversed hazard rate order which in turn implies usual stochasticorder. For a comprehensive discussion on various stochastic orderings, one may refer to Shakedand Shanthikumar (2007) and Muller and Stoyan (2002).We shall use the notion of majorization which is quite useful in establishing variousinequalities. Let x(1) x(n) be the increasing arrangement of the components of themj j n ni=1 x(i) = i=1 y(i). For an extensive andnwith hazard rate = (1, . . . , n) (1, . . . , n) = Xn:n rh Xn:n,vector x = (x1, . . . , xn). The vector x is said to majorize the vector y, written as x y, ifi=1 x(i) i=1 y(i) for j = 1, . . . , n 1, andcomprehensive discussion on the theory and applications of the majorization order, one mayrefer to Marshall and Olkin (1979). Bon and Paltanea (1999) introduced a pre-order on n+,called p-larger order. The vector x in n+ is said to be p-larger than another vector y, writtenpas x y, if i=1 x(i) i=1 y(i) for j = 1, . . . , n. Moreover, it holds thatm px y = x ypfor x, y n+. The converse is, however, not true. For example, we have (2, 7) (3, 5), but themajorization order clearly does not hold between these two vectors.2Main resultIn this section, we stochastically compare two largest order statistics in terms of the re-versed hazard rate order both of which are arising from independent heterogeneous gammasamples.Theorem 1. Let X1, . . . , Xn be independent gamma random variables with Xi having shapeparameter 0 0, the distribution function of Xn:n is given byFXn:n(t) =ni=1i0 (r)ur1eiudu(7)with the reversed hazard raterXn:n(t) =ni=1 t0tr1eitur1eiudu=nt i=1 10eitur1eitudu.It can be seen that rXn:n(t) will be schur convex in (1, . . . , n) if we could prove the ith termin the summation above is convex in it, i = 1, . . . , n (cf. Marshall and Olkin, 1979, p.64).Thus, the problem reduces to prove that the functionf(y) = 10eyxr1eyxdxis convex in y 0. Taking derivative with resect to y, one getsf (y) =ey(100 1),j j(1, . . . , n) (1, . . . , n) = Xn:n rh Xn:n. t r1 xreyxdx 0 xr1eyxdx 1 2xr1eyxdxand taking derivative to f (y), we havesgn=f (y) 1xr1eyxdx 1xr+1eyxdx 1xr1eyxdx0+2xreyxdx 0xr1eyxdx0xreyxdxsgn=0 0r1 yx r+1 yx0 0xreyxdx xreyxdx0 0 x e dx2 2 2 + 1g(y).000xreyxdx.It can be readily veried that the function 1xr+1eyxdx01xreyxdx0in decreasing in y 0, ) and hence reach minimum at origin. Then the condition for theinequality(g(y) 2 2 2 +r +1r +2) 0isr 13 2 2We therefore nish the proof.Remark 2. It is obvious that Theorem 1 strengthen a similar result of Sun and Zhang (2005)from usual stochastic order to reversed hazard rate order. Recently, Khaledi et al. (2011)established a similar result for the scale model under some condition which, however, is notsatised by gamma case discussed here.As an immediate consequence of Theorem 1, we can get the following corollary whichactually gives a lower bound for the reversed hazard rate function of the lifetime of a parallelsystem with heterogeneous gamma components.Corollary 3. Let X1, . . . , Xn be independent gamma random variables with Xi having shapeparameter 0 r 3.8 and scale parameter i, i = 1, . . . , n, and Y1, . . . , Yn be an independentrandom sample from a gamma population with shape parameter r and scale parameter . Then = Xn:n rh Yn:n, 1 1 1 1 1xedxxedx 1 1( ) 1 r+1 yxxreyxdx 1 2 + 2 10xr1eyxdx0 2 3.82843.10.80.6r ( t ;1, 2 ,9 )0.40.2r ( t ; 4 , 4 , 4 )r ( t ; 2 , 3 , 7 )00.511.522.53图 1: Plots of reversed hazard rate functions with shape parameter r = 0.5.10.80.6r ( t ;1, 2 ,9 )0.40.2r ( t ; 4 , 4 , 4 )r ( t ; 2 , 3 , 7 )00.511.522.53图 2: Plots of reversed hazard rate functions with shape parameter r = 3.i=1 i)/n is the arithmetic mean of is.In order to illustrate the result obtained in Theorem 1, we provide the following numericalexample. Let (X1, X2, X3) be a vector of independent heterogeneous gamma random vari-ables with common shape parameter r and scale parameter vector (1, 2, 3). Denote byr(t; 1, 2, 3) the reversed hazard rate function of X3:3. Obviously, it holds thatm mFigure 1 (when r = 0.5) and Figure 2 (when r = 3) present the pictures of the reversed hazardrate functions of X3:3 which is in accordance with the result of Theorem 1.n where = (1, 2, 7) (2, 3, 7) (4, 4, 4).参考文献（References）1 Balakrishnan, N. (2007) Permanents, order statistics, outliers, and robustness. RevistaMatematica Complutense 20, 7-107.2 Balakrishnan, N. and Rao, C. R. (1998a) Handbook of Statistics. Vol. 16: Order Statistics:Theory and Methods. Amsterdam: Elsevier.3 Balakrishnan, N. and Rao, C. R. (1998b) Handbook of Statistics. Vol. 17: Order Statistics:Applications. Amsterdam: Elsevier.4 Boland, P. J., EL-Neweihi, E. and Proschan, F. (1994) Applications of the Hazard RateOrdering in Reliability and Order Statistics. Journal of Applied Probability 31, 180-192.5 Bon, J. L. and Paltanea, E. (1999) Ordering properties of convolutions of exponential ran-dom variables. Lifetime Data Analysis 5, 185-192.6 Bon, J. L. and Paltanea, E. 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