Hazard rate ordering of k-out-of-n systems

Moshe Shaked and Jevaveerasingam Shanthikumar

Statistics & Probability Letters, 1995, vol. 23, issue 1, 1-8

Abstract: Let {X1, X2, ..., Xn} be a set of independent continuous random lifetimes and let {Y1, Y2, ..., Yn} be another set of independent continuous random lifetimes. Let X(1) [less-than-or-equals, slant] X(2) [less-than-or-equals, slant] ... [less-than-or-equals, slant] X(n) be the order statistics associated with the Xi's, and let Y(1) [less-than-or-equals, slant] Y(2) [less-than-or-equals, slant] ... [less-than-or-equals, slant] Y(n) be the order statistics associated with the Yi's. Several recent papers have given conditions under which X(k) [less-than-or-equals, slant]hr Y(k), K = 1, 2, ..., n, where '[less-than-or-equals, slant]hr' denotes the hazard rate order. For example, it is known that if the Xi's are independent and identically distributed, and if the Yi's are independent and identically distributed, and if Xi [less-than-or-equals, slant]hr Yi, I = 1, 2, ... n, then X(k) [less-than-or-equals, slant]hr Y(k). In fact, if the Xi's are not necessarily identically distributed and the Yi's are not necessarily identically distributed, but X[alpha] [less-than-or-equals, slant]hr Y[beta], [alpha], [beta] = 1, 2, ..., n, then it is still true that X(k) [less-than-or-equals, slant]hr Y(k), K = 1, 2, ..., n. The purpose of this paper is to obtain an even stronger result. Our proof is different than the other proofs in the literature and is more intuitive. Some applications in reliability theory are given.

Keywords: Stochastic; ordering; Order; statistics; Series-parallel; and; parallel-series; systems; Hazard; rate; ordering (search for similar items in EconPapers)
Date: 1995
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Citations: View citations in EconPapers (2)

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