Some new results on stochastic comparisons of parallel systems

Abstract

Let X1,…,Xn be independent exponential random variables with Xi having hazard rate . Let Y1,…,Yn be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏i=1nλi)1/n, the geometric mean of the λis. Let Xn:n = max{X1,…,Xn}. It is shown that Xn:n is greater than Yn:n according to dispersive as well as hazard rate orderings. These results lead to a lower bound for the variance of Xn:n and an upper bound on the hazard rate function of Xn:n in terms of . These bounds are sharper than those obtained by Dykstra et al. ((1997), J. Statist. Plann. Inference65, 203–211), which are in terms of the arithmetic mean of the λis. Furthermore, let X1*,…,Xn∗ be another set of independent exponential random variables with Xi∗ having hazard rate λi∗, i = 1,…,n. It is proved that if (logλ1,…,logλn) weakly majorizes (logλ1∗,…,logλn∗, then Xn:n is stochastically greater than Xn:n∗.