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 nn from an exponential distribution with hazard rate λλ, and for 1≤k≤n1≤k≤n, let us denote by Fk:n(λ) the distribution function of the corresponding kthkth order statistic. Let us consider mm random samples of same size nn from exponential distributions having respective hazard rates λ1,…,λmλ1,…,λm. Assume that p1,…,pm>0p1,…,pm>0, such that ∑i=1mpi=1, and let UU and VV be two random variables with the distribution functions Fk:n(λ) and ∑i=1mpiFk:n(λi), respectively. Then, VV is greater in the hazard rate order (or the usual stochastic order) than UU if and only if λ≥∑i=1mpiλikk, and VV is smaller in the hazard rate order (or the usual stochastic order) than UU if and only if λ≤min1≤i≤mλiλ≤min1≤i≤mλi, for all k=1,…,nk=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.