Mathematical Sciences: Statistical Inferences, Decision Theory, and Asymptotics of Eigenvalues and Eigenprojections

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Research is proposed on a variety of problems in statistical decision theory. The main focus of this research is the evaluation of inferential methods with admissibility as one evaluative criterion. The connection between the admissibility of formal Bayes rules and the recurrence of symmetric Markov chains forms an important portion of the theoretical background for this work. In a nonparametric setting an alternative method of inducing inferences is proposed. This alternative is easily implemented numerically and has connections with both frequentist and Bayesian bootstrap procedures. Another area of proposed research concerns questions related to the asymptotic distribution of eigenvalues, eigenprojections, and singular values. Previous work contains new techniques which provide a unified treatment of eigenalues and singular values. Applications of these results to concrete examples and extension of the basic results to eigenprojection problems is proposed. Statistics deals with the activity of making statements about populations on the basis of available samples or data. Since information is most often incomplete, such statements are typically in error. The quantification of this error is a standard statistical activity and is often expressed in terms of probability. The research supported by this grant concerns various methods for quantifying statistical errors and the properties of such methods. The ultimate goal of the research is the development of new statistical methods for obtaining probabilistic quantification of errors in inferential statements.