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Stochastic Ordering of Exponential Family Distributions and Their Mixturesxk

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Yaming Yu
Yaming Yu*
Affiliation:
University of California, Irvine
*
∗Postal address: Department of Statistics, University of California, Irvine, Irvine, CA 92697-1250, USA. Email address: yamingy@uci.edu
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Abstract

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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.

Keywords

Binomial mixtureconvolutiongamma mixturehazard rate orderlikelihood ratio orderlog-concavitynegative binomial mixturePoisson mixturestochastic comparisonstochastic orders

MSC classification

Secondary: 60E15: Inequalities; stochastic orderings 60E05: Distributions
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 1 , March 2009 , pp. 244 - 254
Copyright
Copyright © Applied Probability Trust 2009 

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