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 λ , and for 1 ≤ k ≤ n , let us denote by F k : n ( λ ) the distribution function of the corresponding k t h order statistic. Let us consider m random samples of same size n from exponential distributions having respective hazard rates λ 1 , … , λ m . Assume that p 1 , … , p m > 0 , such that ∑ i = 1 m p i = 1 , and let U and V be two random variables with the distribution functions F k : n ( λ ) and ∑ i = 1 m p i F k : n ( λ i ) , respectively. Then, V is greater in the hazard rate order (or the usual stochastic order) than U if and only if λ ≥ ∑ i = 1 m p i λ i k k , and V is smaller in the hazard rate order (or the usual stochastic order) than U if and only if λ ≤ min 1 ≤ i ≤ m λ i , for all k = 1 , … , n . 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.