Bounds for mixtures of order statistics from exponentials and applications

This paper deals with the stochastic comparison of order statistics and their mixtures. For a random sample of size n from an exponential distribution with hazard rate [lambda], and for 1<=k<=n, let us denote by the distribution function of the corresponding kth order statistic. Let us consider m random samples of same size n from exponential distributions having respective hazard rates [lambda]1,...,[lambda]m. Assume that p1,...,pm>0, such that , and let U and V be two random variables with the distribution functions and , respectively. Then, V is greater in the hazard rate order (or the usual stochastic order) than U if and only if , and V is smaller in the hazard rate order (or the usual stochastic order) than U if and only if [lambda]<=min1<=i<=m[lambda]i, for all k=1,...,n. These properties are used to find the best bounds for the survival functions of order statistics from independent heterogeneous exponential random variables. For the proof, we will use a mixture type representation for the distribution functions of order statistics.