Bayesian nonparametric and semiparametric modeling using Dirichlet process mixing: Full inference with novel applications

Parametric modeling has long dominated both classical and Bayesian inference work. The restrictive assumptions regarding the underlying population(s) forced by common parametric families of distributions necessitate the development of more flexible models. Mixture distributions are very attractive in this regard. Dirichlet process mixed models form a particular class of Bayesian nonparametric mixture models that is becoming increasingly popular. Such models result by assuming a random mixing distribution, taken to be a realization from a Dirichlet process, for the mixture. Fitting of Dirichlet process mixed models is well established in the literature by now. However, inference for population functionals is limited to posterior expectations rendering them useful mainly in semiparametric settings. We first develop a computational method for obtaining the posterior of general functionals associated with the random mixture distribution arising through Dirichlet process mixing. Hence full inference is enabled under either semiparametric or fully nonparametric specifications, as we illustrate throughout the dissertation. The rest of the thesis is devoted to novel modeling approaches employing Dirichlet process mixed models. In the context of median regression modeling, an attractive alternative to mean regression from many respects, we propose and investigate the use of two flexible mixture models for the error distribution achieving increased variability, skewness and general tail behavior. We also consider the problem of nonparametric, or semiparametric modeling and inference for distribution functions subject to probability orderings constraints. Such order restrictions are often desirable in comparing two or more populations. The Bayesian paradigm provides a convenient framework for the development of related modeling since any probability order postulated a priori is preserved to the posterior analysis. Our approach handles the cases of stochastic order, the most common probability ordering, and variability order.