Statistical Inferences and Markov Chains, Admissibility, and Strong Inconsistency

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DMS 96-26601 Eaton This research involves the evaluation of statistical inferences which are expressed as data dependent probability distributions. Inferences of particular interest include both posterior distributions and predictive distributions which are derived from improper prior distributions coupled with the formal application of Bayes Theorem. Questions concerning the admissibility of inferences have lead to the development of new and powerful techniques based on a fundamental connection between issues in statistical decision theory and recurrence/transcience issues involving symmetric Markov Chains. This fundamental connection is further developed in this research along with the somewhat allied noton of strong inconsistency for inferences. Applications of this research include the evaluation and improvement of inferential techniques in high dimensional models such as those which arise in large scale environmental models and those attempting to describe global change. %%% Many problems of statistics, such as those which arise in environmental studies, involve the simultaneous description and/or simultaneous prediction of many variables. In such cases, one often speaks of "high dimensional problems". This research involves the construction, evaluation and comparison of statistical methods which are relevant in these high dimensional problems. Recent developments have shown that the problems which arise in this research are, rather surprisingly, closely connected with certain problems which arise in a seemingly unrelated area of probability. This connection promises to yield exciting new insights regarding both practical and theoretical issues involving high dimensional problems. ***