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

摘要

An exact formula for the convolution of twotdensities 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 aret-densities. Bayesian testing and posterior normalization then require calculation of convolutions oftdensities. 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 oftdensities also arise in the Behrens-Fisher problem and in Bayesian inference concerning the common mean of two samples.