Stochastic Ordering of Exponential Family Distributions and Their Mixtures

TL;DR

This paper studies stochastic orderings among exponential family distributions and their mixtures, providing unified theorems that extend known results for Poisson, binomial, negative binomial, and gamma distributions.

Contribution

It introduces a general theorem based on relative log-concavity that unifies and extends existing stochastic comparison results for various exponential family distributions and their convolutions.

Findings

Unified comparison theorems for exponential family distributions.

Generalization of convolution comparison results for gamma and negative binomial distributions.

Extension of stochastic ordering results to mixtures and convolutions.

Abstract

We investigate stochastic comparisons between exponential family distributions and their mixtures with respect to the usual stochastic order, the hazard rate order, the reversed hazard rate order, and the likelihood ratio order. A general theorem based on the notion of relative log-concavity is shown to unify various specific results for the Poisson, binomial, negative binomial, and gamma distributions in recent literature. By expressing a convolution of gamma distributions with arbitrary scale and shape parameters as a scale mixture of gamma distributions, we obtain comparison theorems concerning such convolutions that generalize some known results. Analogous results on convolutions of negative binomial distributions are also discussed.