Developments and challenges in mixture models

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Mixture models are of considerable value in the statistical analysis of data because they provide a very flexible way of modeling. At the same time, the subject of mixture models is particularly challenging because the traditional theory of parametric inference often can not be applied directly to problems in mixture models. The objective of the proposal is to develop theory and innovative methodology in the field of mixture models and empirical likelihood methods to cope with statistical problems in a variety of applications for which no simple solution exists. This proposal is organized by three projects. In the first project, a novel modeling of regression mixtures with scalar responses and functional predictors will be carried out. The functional principal component techniques will be used to achieve dimension reduction. Order selection in the resulting regression mixture will then be addressed. Mixture models have many natural applications in genetics. In the second project, mixture models will be used to detect allelic association in the presence of locus and /or allelic heterogeneity. Since the mixture model approach takes into account locus heterogeneity, it is expected to have greater power in association analysis than other existing methods. The empirical likelihood, which is a nonparametric method of inference based on a data-driven likelihood ratio function, has become a widely used and studied statistical method over the past fifteen years. In the third project, the empirical likelihood method will be explored to produce efficient inference in missing data problems by using auxiliary information. This research program is novel and multi-faceted. It will draw on several important statistical tools such as mixture models, functional data analysis, and empirical likelihood methods to tackle important scientific questions. It raises several theoretical problems of interest, and in addition, success of this research program will yield significant contribution to the theory and application of mixture models. In turn, these results will be useful in problems arising in many fields in the natural sciences and engineering. The theory and methodologies developed will also have great scope for extensions and applications.