Methods in Monte Carlo Computation, Astrophysical Data Analysis and Hypothesis Testing With Multiply-Imputed Data

Abstract

We present three topics in this thesis: the next generation warp bridge sampling, Bayesian methods for modeling source intensities, and large-sample hypothesis testing procedures in multiple imputation.

Bridge sampling is an effective Monte Carlo method to estimate the ratio of the normalizing constants of two densities. The Monte Carlo errors of the estimator are directly controlled by the overlap between the densities. In Chapter 1, we generalize the warp transformations in Meng and Schilling (2002), and introduce a class of stochastic transformation, called warp-U transformation, which aims at increasing the overlap of the densities of the transformed data without altering the normalizing constants. Warp-U transformation is determined by a Gaussian mixture distribution, which has reasonable amount of overlap with the density of unknown normalizing constant. We show warp-U transformation reduces the f-divergence of two densities, thus bridge sampling with warp-U transformed data has better statistical efficiency than that based on the original data. We then propose a computationally efficient method to find a Gaussian mixture distribution and investigate the performance of the corresponding warp-U bridge sampling. Finally, theoretical and simulation results are provided to shed light on how to choose the tuning parameters in the algorithm.

In Chapter 2, we propose a Bayesian hierarchical model to study the distribution of the X-ray intensities of stellar sources. One novelty of the model is its use of a zero-inflated gamma distribution for the source intensities to reflect the possibility of “dark” sources with practically zero luminosity. To quantify the evidence for “dark” sources, we develop a Bayesian hypothesis testing procedure based on the posterior predictive p-value. Statistical properties of the model and the test are investigated via simulation. Finally, we apply our method to a real dataset from Chandra.

Chapter 3 presents large-sample hypothesis testing procedures in multiple imputation, a common practice to handle missing data. Several procedures are classified, discussed, and compared in details. We also provide an improvement of a Wald-type procedure and investigate a practical issue of the likelihood-ratio based procedure.