Asymptotics of Mixture Model Selection

Abstract

In this thesis, we consider the likelihood ratio test (LRT) when testing for homogeneity in a three component normal mixture model. It is well-known that the LRT in this setting exhibits non-standard asymptotic behaviour, due to non-identifiability of the model parameters and possible degeneracy of Fisher Information matrix. In fact, Liu and Shao (2004) showed that for the test of homogeneity in a two component normal mixture model with a single fixed component, the limiting distribution is an extreme value Gumbel distribution under the null hypothesis, rather than the usual chi-squared distribution in regular parametric models for which the classical Wilks' theorem applies. We wish to generalise this result to a three component normal mixture to show that similar non-standard asymptotics also occurs for this model. Our approach follows closely to that of Bickel and Chernoff (1993), where the relevant asymptotics of the LRT statistic were studied indirectly by first considering a certain Gaussian process associated with the testing problem. The equivalence between the process studied by Bickel and Chernoff (1993) and the LRT was later proved by Liu and Shao (2004). Consequently, they verified that the LRT statistic for this problem diverges to infinity at the rate of loglog n; a statement that was first conjectured in Hartigan (1985). In a similar spirit, we consider the limiting distribution of the supremum of a certain quadratic form. More precisely, the quadratic form we consider is the score statistic for the test for homogeneity in the sub-model where the mean parameters are assumed fixed. The supremum of this quadratic form is shown to have a limiting distribution of extreme value type, again with a divergence rate of loglog n. Finally, we show that the LRT statistic for the three component normal mixture model can be uniformly approximated by this quadratic form, thereby proving that that the two statistics share the same limiting distribution.