Optimal estimation in constrained problems with biostatistical application

我的主要研究兴趣是决策理论中约束参数的最优估计及其在非参数回归、函数和图像估计、基于函数的判别和分类等生物统计问题中的应用。我特别感兴趣的设置（通常称为新古典），其中对象（信号/图像）的数量n很大，训练样本相对较少，即p很小。这样的数据的一个例子来自通过获得时间序列测量的神经退行性疾病的早期进展的研究。Buckerton和Donoho（BD）（1995）给出了一种有效的基于小波的启发式判别分析方法。然而，在这种情况下，在估计未知光滑度函数的情况下，Donoho和Johnstone（1995）表明小波阈值估计量在Besov类的范围内同时接近极小极大。我建议在这里开发一个框架阈值的手段，在判别分析，模仿的功能估计问题，以获得渐近最佳的结果。另一个重要的问题，在这个新古典的设置是渐近最优估计的协方差矩阵。我还建议研究以下类型的函数估计问题，使用小波阈值，以获得接近最优的结果：1。对于离散数据，如在非参数二元或泊松回归中，以及他研究泊松计数数据和泊松回归中的变化的技术，以便研究事件数据，如流行病; 2.在删失回归中：3.对于图像恢复小波估计的基础上收缩和最小化SURE各向异性函数类（也涉及时变数据的功能磁共振成像图像）和在这里的相关数据的变点问题。我也对分类数据的精确推理和决策理论中的问题感兴趣。

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.