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

This article introduces a novel, Bayesian, semi-parametric approach to inference for both elliptical and 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. A feature of our approach is that the partial lack of identifiability inherent in both elliptical and normal/independent distributions can be accounted for by incorporating a restriction on the diagonal elements of the scale matrix. Posterior computation is carried out using a Markov chain Monte Carlo algorithm.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, our methodology is illustrated with both simulated and real data.