Some new results on stochastic comparisons of parallel systems

Abstract

Let X 1 ,…, X n be independent exponential random variables with X i having hazard rate . Let Y 1 ,…, Y n be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏ i =1 n λ i ) 1/ n , the geometric mean of the λ i s. Let X n : n = max{ X 1 ,…, X n }. It is shown that X n : n is greater than Y n : n according to dispersive as well as hazard rate orderings. These results lead to a lower bound for the variance of X n : n and an upper bound on the hazard rate function of X n : n in terms of . These bounds are sharper than those obtained by Dykstra et al. ((1997), J. Statist. Plann. Inference 65, 203–211), which are in terms of the arithmetic mean of the λ i s. Furthermore, let X 1 * ,…, X n ∗ be another set of independent exponential random variables with X i ∗ having hazard rate λ i ∗ , i = 1,…, n . It is proved that if (logλ 1 ,…,logλ n ) weakly majorizes (logλ 1 ∗ ,…,logλ n ∗ , then X n : n is stochastically greater than X n : n ∗ .