Variable Selection in High-Dimensional Modeling and Its Oracle Properties

高维数据，如生物技术和基因数据、金融数据、卫星图像和高光谱图像，在我们的日常生活中是司空见惯的。事实上，高维数据分析已经成为统计学中一个重要的研究课题。变量选择是高维统计建模的基础。目前使用的许多方法都是逐步选择过程，计算量大且忽略了选择过程阶段的随机误差。本研究涉及多种数据分析技术，以开发统一有效的高维统计建模变量选择程序。该项目的目标是显著提高分析复杂高维数据的工具的可用性。在这个项目中，提出了惩罚最小二乘法和惩罚似然方法来为高维数据分析中使用的各种模型选择显著变量。所提出的方法与其他方法不同，因为它通过估计其系数为零来删除不显著的协变量。换句话说，它同时选择重要变量并估计其回归系数，从而使人们能够为估计的参数构建置信区间。提出了一种求解惩罚最小二乘和惩罚似然优化问题的算法。研究并给出了所得估计量的收敛速率和抽样性质。

High-dimensional data, such as, biotech and genetic data, financial data, satellite imagery and hyper-spectral imagery, are commonplace in our daily life. Indeed, high-dimensional data analysis has become an important research topic in statistics. Variable selection is fundamental to high-dimensional statistical modeling. Many approaches currently in use are stepwise selection procedures, which are expensive in computation and ignore stochastic errors in the stage of selection process. This research involves a variety of data-analytic techniques for developing a unified effective variable selection procedure in high-dimensional statistical modeling. The goal of this project is to significantly enhance the availability of tools for analyzing complicated high-dimensional data.In this project, penalized least squares and a penalized likelihood approach are proposed to select significant variables for various models used in high-dimensional data analysis. The proposed approach is distinguished from others since it deletes insignificant covariates by estimating their coefficients to be zero. In the other words, it simultaneously selects significant variables and estimates their regression coefficients, and thereby enables one to construct confidence intervals for the estimated parameters. An algorithm is proposed for finding solutions to optimization problems involved in the penalized least squares and penalized likelihood. The rates of convergence and the sampling properties of the resulting estimators are investigated and presented.