Some new results on stochastic comparisons of parallel systems (Q2725308)

Order statistics play an important role in reliability theory. The time to failure of a k-out-of-n system corresponds to the \((n-k+1)\) th order statistic. \textit{S. Kochar} and \textit{C. Ma} [Stat. Probab. Lett. 43, No. 3, 321-324 (1999; Zbl 0926.62005] studied the problem of stochastically comparing the order statistics of non-identically distributed exponential random variables with those corresponding to independent and identically distributed exponential random variables. In this note, the authors obtain a lower bound for the variance of the maximum of non-identically distributed exponential random variables and an upper bound for the hazard function of this maximum. These bounds are sharper than those obtained by \textit{R. Dykstra} et al. [J. Stat. Plann. Inference, 65, No. 2, 203-211 (1997; Zbl 0915.62044)]. They also prove a result of \textit{G. Pledger} and \textit{F. Proschan} [Optimizing Meth. Statist., Proc. Sympos. Ohio State Univ. 1971, 89-113 (1971; Zbl 0263.62062)] under the p-larger ordering.