Stochastic comparisons of parallel systems of heterogeneous exponential components

Abstract

Let X 1, …, X n be independent exponential random variables with X i having hazard rate λ i , i = 1, …, n. Let λ = ( λ 1, …, λ n ). Let Y 1, …, Y n be a random sample of size n from an exponential distribution with common hazard rate λ = Σ n i = 1 λ i n . The purpose of this paper is to study stochastic comparisons between the largest order statistics X n: n and Y n: n from these two samples. It is proved that the hazard rate of X n: n is smaller than that of Y n: n . This gives a convenient upper bound on the hazard rate of X n: n in terms of that of Y n: n . It is also proved that Y n: n is smaller than X n: n according to dispersive ordering. While it is known that the survival function of X n: n is Schur convex in λ, Boland, El-Neweihi and Proschan [ J. Appl. Prohab. 31 (1994) 180–192] have shown that for n > 2, the hazard rate of X n: n is not Schur concave. It is shown here that, however, the reversed hazard rate of X n: n is Schur convex in λ.