em-Test for Univariate Finite Gaussian Mixture Models

Abstract

While the success of the EM-test in the previous two chapters was confined to finite mixture models with subpopulation distributions belonging to a one-parameter distribution family, the underlying principle of the EM-test is generally applicable. This principle involves comparing the degree of improvement achieved after several EM-iterations starting from a fitted null model. However, this approach does not always yield a clear limiting distribution for the resulting test statistic, which is essential for practical testing procedures. A remarkable revelation occurs when the EM-test is applied to finite Gaussian mixture models. The meticulously designed EM-test for determining the order of the finite Gaussian mixture model leads to elegant and unexpected chi-square limiting distributions. These distributions align with the striking results seen in the likelihood ratio test for regular models. Chapter 15 offers a detailed account of this remarkable achievement.