A Likelihood Ratio Test of a Homoscedastic Multivariate Normal Mixture Against a Heteroscedastic Multivariate Normal Mixture

Abstract

The multivariate finite normal mixture model is one of the most commonly used tools to analyze a heterogeneous data. When using the multivariate finite normal mixture model, one is usually interested in knowing whether a homoscedastic mixture model can be used to simplify the model. The likelihood ratio test (LRT) is the most popular statistic tool to choose between two nested models. Under the null model of a homoscedastic multivariate normal mixture, the asymptotic χ2 distribution is commonly used to approximate the null distribution of the LRT statistic. However, it is demonstrated using numerical studies that the χ2 distribution approximation is not satisfactory and fails to control the nominal type I error unless the sample size is larger than 2000, the mixture components are well-separated, and the singular solutions are avoided. A parametric bootstrap method is further proposed to approximate the distribution of the LRT statistic and its effectiveness is evaluated through extensive numerical studies.