A Bayesian semiparametric Gaussian copula approach to a multivariate normality test

Semiparametric copulas are useful tools for modeling a multivariate distribution whose dependence structure is induced by a known copula and whose marginal distributions are estimated. In this paper, a Bayesian semiparametric copula approach is used to model the underlying multivariate distribution Ftrue. First, the Dirichlet process is constructed on the unknown marginal distributions of Ftrue. Then a Gaussian copula model is utilized to capture the dependence structure of Ftrue. As a result, a Bayesian multivariate normality test is developed by combining the relative belief ratio and the Energy distance. Various interesting theoretical results of the approach are derived. Several examples that cover the high dimensional case are discussed to illustrate the approach.