Stochastic comparisons of parallel systems of heterogeneous exponential components

Abstract Let X1, …, Xn be independent exponential random variables with Xi having hazard rate λi, i = 1, …, n. Let λ = (λ1, …, λn). Let Y1, …, Yn 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 Xn:n and Yn:n from these two samples. It is proved that the hazard rate of Xn:n is smaller than that of Yn:n. This gives a convenient upper bound on the hazard rate of Xn:n in terms of that of Yn:n. It is also proved that Yn:n is smaller than Xn:n according to dispersive ordering. While it is known that the survival function of Xn: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 Xn:n is not Schur concave. It is shown here that, however, the reversed hazard rate of Xn:n is Schur convex in λ.