Bounds for mixtures of order statistics from exponentials and applications
Bounds for mixtures of order statistics from exponentials and applications
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DOI：
10.1016/j.jmva.2011.01.006
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发表时间：
2011-05
影响因子：
1.6
通讯作者：
Eugen Pltnea
中科院分区：
文献类型：
--
作者：
Eugen Pltnea
本文讨论了顺序统计量及其混合的随机比较。对于风险率为λ的指数分布的大小为n的随机样本，对于1≤k≤n，我们用Fk：n(λ)表示相应的第k个顺序统计量的分布函数。让我们考虑来自指数分布的m个相同大小n的随机样本，它们具有各自的风险比率λ1，…，λm.假设p1，…，PM>；0，使得∑i=1mpi=1，并且设U和V是两个随机变量，分别具有分布函数Fk：N(λ)和[公式：见正文]。那么，当且仅当λ≥∑i=1mpiλikk时，在危险率次序(或通常的随机次序)中，V大于U，当且仅当λ≤=1，…时，在危险率次序(或通常的随机次序)中，V小于U这些性质被用来从独立的异类指数随机变量中寻找顺序统计量的生存函数的最优界。为了证明，我们将使用顺序统计量的分布函数的混合型表示。
This paper deals with the stochastic comparison of order statistics and their mixtures. For a random sample of size n from an exponential distribution with hazard rate λ, and for 1≤k≤n, let us denote by Fk:n(λ)the distribution function of the corresponding kth order statistic. Let us consider m random samples of same size n from exponential distributions having respective hazard rates λ1,…,λm. Assume that p1,…,pm>0, such that ∑i=1mpi=1, and let U and V be two random variables with the distribution functions Fk:n(λ)and [Formula: see text] , respectively. Then, V is greater in the hazard rate order (or the usual stochastic order) than U if and only if λ≥∑i=1mpiλikk, and V is smaller in the hazard rate order (or the usual stochastic order) than U if and only if λ≤min1≤i≤mλi, for all k=1,…,n. These properties are used to find the best bounds for the survival functions of order statistics from independent heterogeneous exponential random variables. For the proof, we will use a mixture type representation for the distribution functions of order statistics.