Schur properties of convolutions of exponential and geometric random variables

Abstract

Convolutions of random variables which are either exponential or geometric are studied with respect to majorization of parameter vectors and the likelihood ratio ordering (⩾lr) of random variables. Let Xλ, …, Xλn be independent exponential random variables with respective hazards λi (means 1/λi), i = 1 …, n. Then if λ = (λ1, …, λn) ⩾m (λ1′, …, λn′) = λ′, it follows that Σi = 1n Xλ ⩾lr Σi = 1n Xλ′1. Similarly if Xp1, …, Xpn are independent geometric random variables with respective parameters p1, …, pn, then p = (p1, …, pn) ⩾m(p′1, …, p′n) = p′ or log p = (log p1, …, log pn) ⩾ m (log p1, …, log pn) = log p′ implies Σi = 1n Xpl ⩾ lr Σi = 1n XP′1. Applications of these results are given yielding convenient upper bounds for the hazard rate function of convolutions of exponential (geometric) random variables in terms of those of gamma (negative binomial) distributions. Other applications are also given for a server model, the range of a sample of i.i.d. exponential random variables, and the duration of a multistate component performing in excess of a given level.