Testing the Order of Multivariate Normal Mixture Models

TL;DR

This paper develops likelihood-based tests for determining the number of components in multivariate normal mixture models, providing asymptotic distributions and validating their effectiveness through simulations.

Contribution

It introduces new EM and likelihood ratio tests for homoscedastic and heteroscedastic mixtures, with derived asymptotic distributions and bootstrap validation.

Findings

Tests have good finite sample size properties

Proposed methods show strong power in simulations

Asymptotic distributions are derived for various scenarios

Abstract

Finite mixtures of multivariate normal distributions have been widely used in empirical applications in diverse fields such as statistical genetics and statistical finance. Testing the number of components in multivariate normal mixture models is a long-standing challenge even in the most important case of testing homogeneity. This paper develops likelihood-based tests of the null hypothesis of $M_0$ components against the alternative hypothesis of $M_0 + 1$ components for a general $M_0 \geq 1$. For heteroscedastic normal mixtures, we propose an EM test and derive the asymptotic distribution of the EM test statistic. For homoscedastic normal mixtures, we derive the asymptotic distribution of the likelihood ratio test statistic. We also derive the asymptotic distribution of the likelihood ratio test statistic and EM test statistic under local alternatives and show the validity of parametric bootstrap. The simulations show that the proposed test has good finite sample size and power properties.