Efficiencies of MCMC and nonparametric estimation methods

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In this research, I will focus on several topics concerning Markov chain Monte Carlo (MCMC) algorithms and nonparametric estimation methods. 1. Optimal scaling of high-dimensional MCMC algorithms: Previous analysis in this field has mostly focused upon studying the diffusion limits of various generic random walk Metropolis algorithms where the target distribution typically satisfies some restrictive conditions. This research aims to extend existing results to algorithms with more complex target distributions and proposal densities. This will lead to more implementation guidelines. 2. Applications of MCMC in the area of condensed matter physics: The reptation quantum Monte Carlo (RQMC) algorithm, which is an "approximate" Metropolis-Hastings algorithm, has been a powerful computational tool in condensed matter physics. In practice, however, this algorithm converges slowly to the target distribution once the dimension reaches the thousands, and its failure to give reliable estimates is well documented. The goal of this research is to develop new algorithms by combining various MCMC techniques with the existing RQMC, and study other applications of MCMC algorithms in condensed matter physics. 3. Applications of Bayesian methods and MCMC algorithms in applied health sciences: we perform a longitudinal analysis on the impact of lipids change over time on the risk of coronary heart disease. The approach we use is fully Bayesian, with posterior sampling done by MCMC methods. 4. Nonparametric estimation methods: It is well-known that the efficiencies of many existing nonparametric estimation methods for distribution, quantile and regression are relatively low on the tails of the distribution, particularly for heavy-tailed distributions. In this research, we will study various families of weighted empirical distribution functions, and use them to develop new estimation methods and study their theoretical properties. These methods will be compared to existing methods in the literature and applied to real-life problems.