Person:

Xu, Xiaojin

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.

What does regressing Y on X versus regressing X on Y have to do with MCMC? It turns out that many strategies for speeding up data-augmentation type algorithms can be understood as fostering independence or “de-correlation” between a regression function and the corresponding residual, thereby reducing or even eliminating dependence among MCMC iterates. There are two general classes of algorithms, those corresponding to regressing parameters on augmented data/auxiliary variables and those that operate the other way around. The interweaving strategy (Yu and Meng, 2011, JCGS) provides a general recipe to automatically take advantage of both, and it is the existence of two different types of residuals that makes the interweaving strategy seemingly magical in some cases and promising in general. The concept of residuals—which depends on actual data—also highlights the potential for substantial improvements when data augmentation schemes are allowed to depend on the observed data. At the same time, there is an intriguing phase transition type of phenomenon regarding choosing (partially) residual augmentation schemes, reminding us once more of the prevailing issue of trade-off between robustness and efficiency. This article reports on these latest theoretical investigations (using a class of normal/independence models) and empirical findings (using a posterior sampling for a Probit regression) in the search for effective residual augmentations—and ultimately more MCMC algorithms—that meet the 3-S criterion: simple, stable, and speedy.