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中的生物统计问题中的应用。我特别感兴趣的是这样的设置(通常称为新古典主义)，其中对象(信号/图像)的数量n大并且存在相对较少的训练样本，即，p是小的。这类数据的一个例子是通过获得测量的时间序列来研究神经退行性疾病的早期进展。Buckheit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，在估计未知光滑性的函数的情况下，Donoho和Johnstone(1995)证明了小波阈值估计量在一定范围的Besov类上同时是近极小极大估计。在这里，我建议开发一个框架，用于对判别分析中的平均值进行阈值设置，以模拟函数估计问题，以获得渐近最优的结果。在这种新古典背景下的另一个重要问题是协方差矩阵的渐近最优估计。我还建议使用小波阈值来研究以下类型的函数估计问题，以获得接近最优的结果：1.对于离散数据，如在非参数二进制或泊松回归中，利用他的技术来研究泊松计数数据和泊松回归中的变化，以便研究事件数据，例如流行病；2.在删失回归中：3.基于各向异性函数类(也涉及功能磁共振图像的时变数据)的小波估计器的图像恢复以及相关数据的变点问题。我还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.