Optimal estimation in constrained problems with biostatistical application

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My main research interest is the optimal estimation of constrained parameters in decision theory and its application to biostatistical problems in non-parametric regression, function and image estimation, discrimination and classification based on functions. I am particularly interested in the setting (often called neo-classical) where n the number of the objects (signals/images) is large and there are relatively few training samples, that is, p is small. An example of such data comes from the study of the early progression of neurodegenerative disorders by obtaining time series of measurements. Buckheit and Donoho (BD) (1995) gave an e®ective wavelet based heuristic for discriminant analysis. In this setting, however, in the case of estimating a function of unknown smoothness Donoho and Johnstone (1995) showed wavelet thresholded estimators are near-minimax simultaneously over a range of Besov classes. I propose here to develop a framework for thresholding the means in a discriminant analysis which mimics the function estimation problem in order to obtain asymptotically optimal results. Another important problem in this neo-classical setting is the asymptotically optimal estimation of the covariance matrix. I also propose to study the following types of function estimation problems using wavelet thresholding to obtain near optimalresults: 1. for discrete data as in non-parametric binary or Poisson regression and to his technique to study changes in Poisson count data and in Poisson regression in order to study event data such as epidemics; 2. in censored regression: 3. for image recovery by wavelet estimators based on shrinkage and minimizing SURE over anisotropic function classes (also involving time-varying data for fMRI images) and in the change point problem here for correlated data. I am also interested in exact inference for categorical data and in problems in decision theory.