New Developments on Bayesian Bootstrap for Unrestricted and Restricted Distributions

Abstract

The recent popularity of Bayesian inference is due to the practical advantages of the Bayesian approach. The Bayesian analysis makes it possible to reflect ones prior beliefs into the analysis. In this thesis, we explore some asymptotic results in Bayesian nonparametric inference for restricted and unrestricted space of distributions. This thesis is divided into two parts. In the first part, we employ the Dirichlet process in a hypothesis testing framework to propose a Bayesian nonparametric chi-squared goodness-of-fit test. Our suggested method corresponds to Lo's Bayesian bootstrap procedure for chi-squared goodness-of-fit test. Indeed, our bootstrap rectifies some shortcomings of regular bootstrap which only counts number of observations falling in each bin in contingency tables. We consider the Dirichlet process as the prior for the distribution of data and carry out the test based on the Kullback-Leibler distance between the Dirichlet process posterior and the hypothesized distribution. We prove that this distance asymptotically converges to the same chi-squared distribution as the classical frequentist's chi-squared test. Moreover, the results are generalized to the chi-squared test of independence for contingency tables. In the second part, our main focus is on Bayesian nonparametric inference for a restricted group of distributions called spherically symmetric distributions. We describe a Bayesian nonparametric approach to perform an inference for a bivariate spherically symmetric distribution. We place a Dirichlet invariant process prior on the set of all bivariate spherically symmetric distributions and derive the Dirichlet invariant process posterior. Indeed, our approach is an extension of the Dirichlet invariant process for the symmetric distributions on the real line to bivariate spherically symmetric distribution where the underlying distribution is invariant under a finite group of rotations. Further, we obtain the Dirichlet invariant process posterior for the infinite transformation group and we prove that it approaches a certain Dirichlet process. Finally, we develop our approach to obtain the Bayesian nonparametric posterior distribution for functionals of the distribution's support when the support satisfies certain symmetry conditions. When symmetry holds with respect to the parallel lines of axes (for example, in two dimensional space x = a and y = b) we employ our approach to approximate the distribution of certain functionals such as area and perimeter for the support of the distribution. This suggests a Bayesian nonparametric bootstrapping scheme. The estimates can be derived based on posterior averaging. Then, our simulation results demonstrate that our suggested bootstrapping technique improves the accuracy of the estimates.