A Concept of Positive Dependence for Exchangeable Random Variables

Abstract

An $n$-variate distribution function is said to be positive dependent by mixture (PDM) if it is a mixture of independent $n$-variate distributions with equal marginals. PDM distributions arise in various contexts of reliability and other areas of statistics. We give a necessary and sufficient condition, by means of independent random variables, for an $n$-variate distribution function to be PDM. The distributions and the expectations of the order statistics of PDM and of independent $n$-variate distributions which have the same marginals, are compared and the results applied to obtain bounds for the reliability of certain "$k$ out of $n$" systems. A characterization of vectors of expectations of order statistics of PDM distribution is shown. Surprisingly many exchangeable distributions are found to be PDM. We prove a closure property of the class of PDM distributions and list some examples.