Mathematical Sciences: Statistical Inference for General Models

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This work consists of research in mathematical statistics. The problems involve inference for spatial processes including the properties of predictors based on estimated covariance functions and prediction problems for non-Gaussian spatial processes. The relationship between prediction and estimation for general stochastic processes will be studied. Researchers will examine the ways a regression interpretation of James-Stein shrinkage estimators permits a unified treatment of a broad class of problems in cluster analysis from a decision theoretic perspective. Previous historical investigations will be extended to cover main themes in the first half of this century. The asymptotic theory of estimation will be studied with particular attention to the behavior of maximum likelihood estimators on infinite dimensional parameter spaces. The application of algorithms for the computation of posterior distributions to problems of image reconstruction will be examined with particular attention to images generated by positron emission tomography. The computation of probability distributions over graphical structures will be investigated with applications to genetics.