Variable Selection in High-Dimensional Modeling and Its Oracle Properties

高维数据，如生物技术和遗传数据、金融数据、卫星图像和超光谱图像，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的一个重要研究课题。 变量选择是高维统计建模的基础。 目前使用的许多方法是逐步选择过程，这是昂贵的计算和忽略随机误差的选择过程中的阶段。本研究涉及各种数据分析技术，以开发一个统一的有效的变量选择程序，在高维统计建模。本计画的目标是大幅提升分析复杂高维资料的工具的可用性，在本计画中，提出惩罚最小二乘法与惩罚似然法，以选择高维资料分析中各种模型的重要变数。所提出的方法是有别于其他，因为它删除了不重要的协变量估计其系数为零。 换句话说，它同时选择重要变量并估计其回归系数，从而使人们能够构建估计参数的置信区间。 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。的收敛速度和抽样特性的估计进行了调查和介绍。

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.