SOME ADVANCES IN BAYESIAN NONPARAMETRIC MODELING

Statistics) SOME ADVANCES IN BAYESIAN NONPARAMETRIC MODELING by Abel Rodŕıguez Institute of Statistics and Decision Sciences Duke University Date: Approved: Dr. Alan E. Gelfand, Supervisor Dr. David B. Dunson, Supervisor Dr. Mike West Dr. Robert Wolpert An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Institute of Statistics and Decision Sciences in the Graduate School of Duke University 2007 Copyright c © 2007 by Abel Rodŕıguez All rights reserved Abstract Many modern scientific problems involve outcomes that are complex, infinite dimensional objects like curves and distributions. Often, no satisfactory statistical methodology is available for inference in these types of problems. Recent research in Bayesian nonparametric methods has focused on extending existing models to accommodate simultaneous inferences for multiple dependent distributions. This dissertation focuses on problems of density estimation on collections of distributions using extensions of the Dirichlet processes, as well as their application to nonparametric regression. The dissertation can be broadly divided in three semi-autonomous pieces. In the first part, comprising chapters 2 to 4, we develop models for the joint estimation of collections of densities in two specific contexts: 1) multicenter studies, where distributions are assumed to form clusters indicating common underlying characteristics and 2) time series where distributions evolve in discrete-time. We demonstrate the versatility of the models through applications in epidemiology, public health and finance. In the second part, which involves chapter 5 and 6, we frame nonparametric regression as a density estimation problem. First, we show that consistency of the density estimates automatically induces pointwise consistency of the functional estimates. From there, we develop methods for functional data analysis based on dependent Dirichlet processes. Specifically, we discuss applications to functional clustering and functional spatial data analysis. Examples of these methods are drawn from oceanography and public health. Finally, chapter 7 introduces a novel nonparametric prior on the space of stochastic processes that provides a flexible alternative to the Gaussian process. This class of models has few precedents in the literature and is different from the models for ivMany modern scientific problems involve outcomes that are complex, infinite dimensional objects like curves and distributions. Often, no satisfactory statistical methodology is available for inference in these types of problems. Recent research in Bayesian nonparametric methods has focused on extending existing models to accommodate simultaneous inferences for multiple dependent distributions. This dissertation focuses on problems of density estimation on collections of distributions using extensions of the Dirichlet processes, as well as their application to nonparametric regression. The dissertation can be broadly divided in three semi-autonomous pieces. In the first part, comprising chapters 2 to 4, we develop models for the joint estimation of collections of densities in two specific contexts: 1) multicenter studies, where distributions are assumed to form clusters indicating common underlying characteristics and 2) time series where distributions evolve in discrete-time. We demonstrate the versatility of the models through applications in epidemiology, public health and finance. In the second part, which involves chapter 5 and 6, we frame nonparametric regression as a density estimation problem. First, we show that consistency of the density estimates automatically induces pointwise consistency of the functional estimates. From there, we develop methods for functional data analysis based on dependent Dirichlet processes. Specifically, we discuss applications to functional clustering and functional spatial data analysis. Examples of these methods are drawn from oceanography and public health. Finally, chapter 7 introduces a novel nonparametric prior on the space of stochastic processes that provides a flexible alternative to the Gaussian process. This class of models has few precedents in the literature and is different from the models for iv collection of distributions that we developed in the first part of the dissertation. As an application, we discuss a stochastic volatility model for option pricing.