Bayesian test of normality versus a Dirichlet process mixture alternative

TL;DR

This paper introduces a Bayesian normality test using Dirichlet process mixture models as alternatives, employing an efficient sampler to detect deviations from normality in univariate and multivariate data.

Contribution

It develops a novel Bayesian test framework with a scalar parameterization and an efficient sampler, improving detection of non-normality without bias towards the alternative.

Findings

Test effectively detects non-normality in simulations.

Method does not favor nonparametric alternatives when data is normal.

Provides a flexible Bayesian approach for normality testing.

Abstract

We propose a Bayesian test of normality for univariate or multivariate data against alternative nonparametric models characterized by Dirichlet process mixture distributions. The alternative models are based on the principles of embedding and predictive matching. They can be interpreted to offer random granulation of a normal distribution into a mixture of normals with mixture components occupying a smaller volume the farther they are from the distribution center. A scalar parametrization based on latent clustering is used to cover an entire spectrum of separation between the normal distributions and the alternative models. An efficient sequential importance sampler is developed to calculate Bayes factors. Simulations indicate the proposed test can detect non-normality without favoring the nonparametric alternative when normality holds.