Bayesian inference on mixture models and their applications

Mixture models are useful in describing a wide variety of random phenomena becauseof their flexibility in modeling. They have continued to receive increasing attentionover the years from both a practical and theoretical point of view. In their applications,estimating the number of mixture components is often the main researchobjective or the first step toward it. Estimation of the number of mixture componentsheavily depends on the underlying distribution. As an extension of normalmixture models, we introduce a skew-normal mixture model and adapt the reversiblejump Markov chain Monte Carlo algorithm to estimate the number of componentswith some applications to biological data.The reversible jump algorithm is also applied to the Cox proportional hazardmodel with frailty. We consider a regression model for the variance components inthe proportional hazards frailty model. We propose a Bayesian model averaging procedurewith a reversible jump Markov chain Monte Carlo step which selects the modelautomatically. The resulting regression coefficient estimates ignore the model uncertaintyfrom the frailty distribution. Finally, the proposed model and the estimationprocedure are illustrated with simulated example and real data.