Exact Methods in Statistical Inference
Seeking exact methods for statistical inference problems is a fundamental and central topic in statistics. Exact methods refer to inference procedures that are able to accurately quantify the uncertainty associated with the statistical model for any finite sample size, in contrast to approximate methods that typically rely on large-sample asymptotic results. In this dissertation, we investigate three popular models that are widely used in practice but with very few exact inference results, which lead to three parts of this dissertation. In the first part, we revisit a classical mean comparison model for multivariate data, also known as the multivariate Behrens-Fisher problem. Specifically, we are interested in testing the mean difference between two multivariate normal samples with unknown covariance structures. Compared with most of the existing methods that only provide approximate p-values, we have derived finite-sample bounds for the null distribution of the test statistic, thus leading to guaranteed control of the Type I error. This is the first general result for exact inference based on the multivariate Behrens-Fisher test statistic. In the second part, we further extend the model to functional data, which are data viewed as functions or curves that are essentially infinite-dimensional. Functional data have become more and more prevalent with the advancement of modern data collection technologies. However, the exact inference for functional mean comparison is formidable in the conventional frequentist framework, so we develop a testing procedure under the generalized p-value framework (Tsui and Weerahandi, 1989), which is effective in dealing with nuisance parameters present in the model. Simulation results and real data analysis indicate that the proposed generalized p-value for functional mean comparison has superb performance in terms of both the size and power of the test. Lastly, we consider the exact inference of a class of Bayesian models in which only partial prior information is available, which is referred to as the Partial Bayes (PB) problem in this dissertation. PB problems arise when data analysts have some type of prior knowledge that is however insufficient to form a known distribution. Historically, such problems were typically handled by the Empirical Bayes (EB) methods. However, EB in general underestimates the model uncertainty, which usually leads to unreliable inference results. In this dissertation, we develop a general framework for studying PB problems using the Inferential Model theory (Martin and Liu, 2013), and obtain interval estimators that are both valid and efficient under mild conditions.
Wang, Purdue University.
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