Mathematical Sciences: Statistifcal Theory and Methodology

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The general goal of this project is the development of statistical theory that is likely to be of direct use to statistical practitioners. Specific projects are developed around the theme of computer-intensive statistical inference. Special topics include bootstrap, jackknife and related resampling methods, permutation tests and group theoretic testing procedures, Gibbs sampling and Markov chains. All these topics concern data mappings of special interest to statisticians. Historically, permutations were the first such mappings studied. Methods like the bootstrap generalize permutations to wider classes of statistical problems, at the expense of less mathematical neatness and greater computational expense. This project addresses ways to overcome these obstacles. Statistical inference is the science of generalization from observed data, often just a small amount, to the larger population from which the data were drawn. The basic principles of statistical inference were laid down before World War II. The basic theory was built around classical mathematics and the bell- shaped curve. Computer-intensive methods, developed in the last 15 years, aim to extend the range, ease, and plausibility of statistical arguments. They substitute specially-developed computer algorithms of classical mathematical approximations relating to the bell-shaped curve. Our research aims to justify these algorithms. Mathematical and computational arguments are used to show that the algorithms automatically produce the classical answers when such answers exist, and still give good answers in the much more common circumstance of problems having no classical solution.