Nonparametric Bayesian inference with single and multivariate random probability measures; heavy tailed time series.

点击翻译按钮获取中文摘要

My research concentrates largely on solving problems based on Bayesian modeling. The ability to perform the computations for Bayesian analysis was made possible by advances in the 1990's. Use of hierarchical Bayesian modeling had almost no chance to be carried out without the powerful computation tools available today. The term hierarchical modeling often refers to the idea that the prior can be split up into further hierarchy layers. Advancement in the speed of computers led to a paradigm shift in using both Bayesian and Frequentist inference. ******The applications of Bayesian methods are motivated by applications in different areas of Statistics, Computer Sciences, and in general Data Science. Machine learning theory, which makes extensive***use of Bayesian Statistics, is an important area in Data Science. My proposed research program focuses mainly on problems in Bayesian nonparametric inference, meaning that only minor restrictions are assumed about the parameters in a population. Adding few restrictions will be necessary in Bayesian Statistics, despite the fact that making minimal assumptions on population is an influential factor to choose nonparametric priors. An example of these assumptions is the symmetry of innovations in general linear models. My proposed research will also study the impact of certain restrictions on the population to develop proper Bayesian nonparametric priors in making inferences.******Work on nonparametric Bayesian methods concentrates on applications of objects similar to the Dirichlet process priors, namely nonparametric priors. The goal of this research is to advance the knowledge both in theory and application. The theoretical work will develop ways to use the nonparametric priors as a fully Bayesian approach to problems such as estimation and testing statistical hypothesis. This research will also consider the large sample behavior for the use of these nonparametric priors. The large sample investigation derives statistical results when the number of observations grow.******The proposed research will also consider the large sample theory in estimating parameters in time series when variables may fluctuate wildly. These variables are encountered frequently in applications in finance, insurance and environmental studies, as models for perturbations that exhibit extreme behavior. I am especially interested in time series that behave like a random walk, where the size of each step shows extreme behaviors. The machinery used in large sample theory of these variables can be used in other research areas such as estimating the support of an unknown population.******Furthermore, identifying the gaps between theoretical techniques and practice will drive new areas of statistical research and important in progress of Statistical sciences. In this research, efforts will be made to progress on this front and to find ways to show practitioners how to apply techniques in real life.*****