Bounds for mixtures of order statistics from exponentials and applications

AbstractThis 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 Fk:n(λ) 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 λ1,…,λm. Assume that p1,…,pm>0, such that ∑i=1mpi=1, and let U and V be two random variables with the distribution functions Fk:n(λ) and ∑i=1mpiFk:n(λi), respectively. Then, V is greater in the hazard rate order (or the usual stochastic order) than U if and only if λ≥∑i=1mpiλikk, and V is smaller in the hazard rate order (or the usual stochastic order) than U if and only if λ≤min1≤i≤mλi, 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

Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables

Let X1,...,Xn be independent exponential random variables with respective hazard rates [lambda]1,...,[lambda]n, and let Y1,...,Yn be independent exponential random variables with common hazard rate [lambda]. This paper proves that X2:n, the second order statistic of X1,...,Xn, is larger than Y2:n, the second order statistic of Y1,...,Yn, in terms of the likelihood ratio order if and only if with . Also, it is shown that X2:n is smaller than Y2:n in terms of the likelihood ratio order if and only if These results form nice extensions of those on the hazard rate order in Pa[caron]lta[caron]nea [E. Pa[caron]lta[caron]nea, On the comparison in hazard rate ordering of fail-safe systems, Journal of Statistical Planning and Inference 138 (2008) 1993-1997].primary, 60E15 secondary, 60K10 Majorization order Weakly majorization order p-larger order Hazard rate order

Some new results on convolutions of heterogeneous gamma random variables

AbstractConvolutions of independent random variables often arise in a natural way in many applied areas. In this paper, we study various stochastic orderings of convolutions of heterogeneous gamma random variables in terms of the majorization order [p-larger order, reciprocal majorization order] of parameter vectors and the likelihood ratio order [dispersive order, hazard rate order, star order, right spread order, mean residual life order] between convolutions of two heterogeneous gamma sets of variables wherein they have both differing scale parameters and differing shape parameters. The results established in this paper strengthen and generalize those known in the literature