  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.
