A Necessary Bayesian Nonparametric Test for Assessing Multivariate Normality

A novel Bayesian nonparametric test for assessing multivariate normal models is presented. Although there are extensive frequentist and graphical methods for testing multivariate normality, it is challenging to find Bayesian counterparts. The approach considered in this paper is based on the Dirichlet process and the squared radii of observations. Specifically, the squared radii are employed to transform the m-variate problem into a univariate problem by relying on the fact that if a random sample is coming from a multivariate normal distribution then the square radii follow a particular beta distribution. While the Dirichlet process is used as a prior on the distribution of the square radii, the concentration of the distribution of the Anderson-Darling distance between the posterior process and the beta distribution is compared to that between the prior process and beta distribution via a relative belief ratio. Key results of the approach are derived. The procedure is illustrated through several examples, in which it shows excellent performance.