Some new results on stochastic comparisons of parallel systems

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 ∗.