Abstract: In this paper, we discuss a novel class of skewed multivariate distributions and, more generally, a method of building such a class on the basis of univariate skewed distributions. The method is based on a general linear transformation of a multidimensional random variable with independent components, each with a skewed distribution. The proposed class of multivariate skewed distributions has a simple form for the pdf, and moment existence only depends on that of the underlying symmetric univariate distributions. In addition, we can freely allow for any mean and covariance structure in combination with any magnitude and direction of skewness. In order to deal with both skewness and fat tails, we introduce multivariate skewed regression models with fat tails based on Student distributions. We present two main classes of such distributions, one of which is novel even under symmetry. Under standard non-informative priors on both regression and scale parameters, we derive conditions for propriety of the posterior and for existence of posterior moments. We describe MCMC samplers for Bayesian inference and analyse an application to biomedical data.
Key words and phrases: Asymmetric distributions, Bayesian inference, heavy tails, Mardia's measure of skewness, orthogonal matrices, posterior propriety.