Stochastic order of sample range from heterogeneous exponential random variables

Peng Zhao, Xiaohu Li

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

Let X1,...,Xn be independent exponential random variables with their respective hazard rates 1,...,n, and let Y1,...,Yn be independent exponential random variables with common hazard rate . Denote by Xn:n, Yn:n and X 1:n, Y1:n the corresponding maximum and minimum order statistics. Xn:n-X1:n is proved to be larger than Y n:n-Y1:n according to the usual stochastic order if and only if λ ≥ (λ̄-1||n i=1λi)1/(n-1) with λ̄ = ∑ni=1λi}/n. Further, this usual stochastic order is strengthened to the hazard rate order for n=2. However, a counterexample reveals that this can be strengthened neither to the hazard rate order nor to the reversed hazard rate order in the general case. The main result substantially improves those related ones obtained in Kochar and Rojo and Khaledi and Kochar.

Original languageEnglish
Pages (from-to)17-29
Number of pages13
JournalProbability in the Engineering and Informational Sciences
Volume23
Issue number1
DOIs
StatePublished - Jan 2009

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