Methods in Hypothesis Testing, Markov Chain Monte Carlo and Neuroimaging Data Analysis

Abstract

This thesis presents three distinct topics: a modified K-S test for autocorrelated data, improving MCMC convergence rate with residual augmentations, and resting state fMRI data analysis. In Chapter 1, we present a modified K-S test to adjust for sample autocorrelation. We first demonstrate that the original K-S test does not have the nominal type one error rate when applied to autocorrelated samples. Then the notion of mixing conditions and Billingsley's theorem are reviewed. Based on these results, we suggest an effective sample size formula to adjust sample autocorrelation. Extensive simulation studies are presented to demonstrate that this modified K-S test has the nominal type one error as well as reasonable power for various autocorrelated samples. An application to an fMRI data set is presented in the end. In Chapter 2 of this thesis, we present the work on MCMC sampling. Inspired by a toy example of random effect model, we find there are two ways to boost the efficiency of MCMC algorithms: direct and indirect residual augmentations. We first report theoretical investigations under a class of normal/independece models, where we find an intriguing phase transition type of phenomenon. Then we present an application of the direct residual augmentations to the probit regression, where we also include a numerical comparison with other existing algorithms.