Some new results on stochastic comparisons of parallel systems
Article
Khaledi, BE, Kochar, S. (2000). Some new results on stochastic comparisons of parallel systems
. JOURNAL OF APPLIED PROBABILITY, 37(4), 1123-1128. 10.1017/S0021900200018301
Khaledi, BE, Kochar, S. (2000). Some new results on stochastic comparisons of parallel systems
. JOURNAL OF APPLIED PROBABILITY, 37(4), 1123-1128. 10.1017/S0021900200018301
Let X1 , . . . , Xn be independent exponential random variables with Xi having hazard rate λi, i = l , . . . , n. Let Y1 , . . . , Yn be a random sample of size n from an exponential distribution with common hazard rate λ̃ = (∏ni=l λi)l/n, the geometric mean of the λis. Let Xn:n = max{Xl , . . . , 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 λ̃. 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 λis. Furthermore, let X*l , . . . , X*n be another set of independent exponential random variables with X*i having hazard rate λ*i, i = \ . . . . n. It is proved that if (log λl , . . . , log λn) weakly majorizes (log λ*l , . . . , log λ*n), then Xn:n is stochastically greater than X*n:n.
