Some research problems under finite mixture models

This thesis studies two types of research problems under finite mixture models. The first type is mixing distribution estimation. It is well-known that the maximum likelihood estimator (MLE) fails under some finite mixture models because their likelihood function is unbounded. This unboundedness occurs, for instance, under the finite normal mixture model, the finite gamma mixture model, and the finite location-scale mixture model. In the literature, different estimation methods have been developed by modifying the likelihood functions to restore consistency. The penalized MLE is one of the popular remedies. Though the consistency of penalized MLE has been studied extensively by many researchers, the results were often acquired under finite mixture models with some specific forms of kernel distribution. We provide a route to establish the consistency of penalized MLE under a unified framework that covers many finite mixture models. This route summarizes and helps to improve the existing results. We also study a novel way to modify the likelihood function based on data augmentation. This modified likelihood function produces a consistent estimator, the augmented MLE, under those finite mixture models with unbounded likelihood functions. In some circumstances, the augmented MLE is more efficient than its competitors in the literature. The second type of research problem is hypothesis testing for homogeneity un- der finite mixture models. We develop two tests for this purpose. First, our work migrates the Expectation-Maximization (EM) test to a finite vector-parameter mixture model with structural parameters. The EM test is a numeric and analytically tractable likelihood-based test, which has been applied to many finite mixture models. The second test is Neyman’s C(α) test, a score test variant. We generalize this test to finite vector-parameter mixture models following the principles of Neyman. Our generalization aligns with some existing results in the literature, though they are motivated differently. Yet we find this generalized test can be asymptotically biased under the finite gamma mixture model. To overcome this deficiency, we develop another C(α) test that we conjecture to be asymptotically unbiased.

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