Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability

AbstractThe basic result of the paper states: Let F1, …, Fn, F1′,…, Fn′ have proportional hazard functions with λ1 ,…, λn , λ1′ ,…, λn′ as the constants of proportionality. Let X(1) ≤ … ≤ X(n) (X(1)′ ≤ … ≤ X(n)′) be the order statistics in a sample of size n from the heterogeneous populations {F1 ,…, Fn}({F1′ ,…, Fn′}). Then (λ1 ,…, λn) majorizes (λ1′ ,…, λn′) implies that (X(1) ,…, X(n)) is stochastically larger than (X(1)′ ,…, X(n)′). Earlier results stochastically comparing individual order statistics are shown to be special cases. Applications of the main result are made in the study of the robustness of standard estimates of the failure rate of the exponential distribution, when observations actually come from a set of heterogeneous exponential distributions. Further applications are made to the comparisons of linear combinations of Weibull random variables and of binomial random variables