Exact convolution oftdistributions, with applications to Bayesian inference for a normal mean withtprior distributions∗

Abstract

An exact formula for the convolution of two t densities with odd degrees of freedom is derived. Such convolutions are of great importance to statistics. For instance, for basic normal inference problems concerning the mean when the variance is unknown, Bayesian analyses tend to end up dealing with marginal likelihood functions for the mean that are t-densities. Bayesian testing and posterior normalization then require calculation of convolutions of t densities. This, as well as the posterior mean and variance, are evaluated in closed form when the sample size is even and the prior has odd degrees of freedom. Convolutions of t densities also arise in the Behrens-Fisher problem and in Bayesian inference concerning the common mean of two samples.