Hypothesis test for normal mixture models: The EM approach

TL;DR

This paper introduces an EM-test for homogeneity in normal mixture models, effectively addressing challenges like unbounded likelihoods and non-identifiability, with theoretical distribution results and practical applications demonstrated through simulations and genetic data examples.

Contribution

It develops an EM-test for normal mixture models that overcomes key technical challenges and derives its limiting distribution under various conditions.

Findings

The EM-test's limiting distribution is a mixture of chi-square distributions.

Simulations show the distribution approximates finite sample behavior well.

The method is demonstrated with genetic data examples.

Abstract

Normal mixture distributions are arguably the most important mixture models, and also the most technically challenging. The likelihood function of the normal mixture model is unbounded based on a set of random samples, unless an artificial bound is placed on its component variance parameter. Moreover, the model is not strongly identifiable so it is hard to differentiate between over dispersion caused by the presence of a mixture and that caused by a large variance, and it has infinite Fisher information with respect to mixing proportions. There has been extensive research on finite normal mixture models, but much of it addresses merely consistency of the point estimation or useful practical procedures, and many results require undesirable restrictions on the parameter space. We show that an EM-test for homogeneity is effective at overcoming many challenges in the context of finite normal mixtures. We find that the limiting distribution of the EM-test is a simple function of the $0.5\chi^2_0+0.5\chi^2_1$ and $\chi^2_1$ distributions when the mixing variances are equal but unknown and the $\chi^2_2$ when variances are unequal and unknown. Simulations show that the limiting distributions approximate the finite sample distribution satisfactorily. Two genetic examples are used to illustrate the application of the EM-test.