Semiparametric models have played an increasingly important role in statistical research and received much attention in both frequentist and Bayesian contexts. They are known to be very flexible while overcoming the problem of ‘curse of dimensionality’, and thus find numerous applications in the fields of econometrics, bioinformatics, biomedicine and others. Therefore, it is an interesting but challenging problem to develop semiparametric models for various circumstances with efficient algorithms for statistical inference. In this thesis, we propose Bayesian approaches for two popular classes of semiparametric models, single-index models for Tobit quantile regression and partially linear additive models with automatic and simultaneous model selection and estimation. Based on Markov Chain Monte Carlo method and mean field variational Bayes approximation scheme, we develop efficient algorithms for posterior inferences. Our approaches extend the scope of the applicabilities of the aforementioned semiparametric models from both theoretical and empirical perspectives. With extensive simulation studies, real data examples and comparative works, the proposed approaches are well demonstrated and illustrated.

​Doctor of Philosophy (SPMS)