Bayesian semi-parametric approaches to normal/independent and elliptical distributions

Abstract

This article introduces new Bayesian, semi-parametric approaches to inference for the general class of elliptical distributions, which includes the normal/independent distributions. The location and scale parameters are modelled parametrically and a suitable transformation of the modular variable is modelled using Dirichlet process mixtures. Posterior computation is carried out using a Metropolis within Gibbs algorithm. A feature of the method is that the partial lack of identifiability inherent in elliptical distributions can be accounted for by incorporating a restriction on the diagonal elements of the scale matrix. A novel technique for model selection, based on an approximation of the deviation information criterion, is introduced. As shown by a numerical study based on simulation, the approach can be used to discriminate between elliptical, and normal/independent distributions. Finally, the approach is illustrated with a real data set and a regression example.