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Abstract: It is well-known that the 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. I 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.
Hypothesis testing for homogeneity under finite mixture models assesses the hypotheses that data are collected from a non-mixture distribution (Null) or a two-subpopulation mixture distribution. I develop an EM test and a C(α) test under finite vector-parameter mixture models for this purpose.