Asymptotic relationships between the D-test and likelihood ratio-type tests for homogeneity

The D-test for homogeneity in finite mixtures is appealing because the D-test statistic depends on the data solely through parameter estimates, whereas likelihood ratio-type test statistics require both parameter estimates and the full data set. In this paper we establish asymptotic equivalences between the D-test and three likelihood ratio-type tests for homogeneity. The first two equivalences, under maximum likelihood and Bayesian estimation frameworks respectively, apply to mixtures from a one-dimensional exponential family; the second equivalence yields a simple limiting null distribution for the D-test statistic as well as a simple limiting distribution under contiguous local alternatives, revealing that the D-test is asymptotically locally most powerful. The third equivalence, under an empirical Bayesian estimation framework, pertains to mixtures from a normal location family with unknown structural parameter; the third equivalence also yields a simple limiting null distribution for the D-test statistic. Simulation studies are provided to investigate finite-sample accuracy of critical values based on the limiting null distributions and to compare the D-test to its competitors regarding power to detect heterogeneity. We conclude with an application to medical data and a discussion emphasizing computational advantages of the D-test.