稳健且有效的回归和变量选择方法研究

如何构造高稳健又高有效的回归估计，尤其在超高维数据中，是一个具有挑战性的工作。对于回归模型和(超)高维环境中的大多数M-估计，虽然在模拟实验中能够验证它们具有一定的稳健性，但是它们的一个重要的稳健性度量- - -有限样本的崩溃点很低，渐进为0。基于分步M-估计的MM估计和ARETE等估计可以改进这些不足，使得它们能够同时具有高稳健性和高有效性，但它们都依赖于一个高稳健的初始估计。也因为这种依赖性，ARETE虽然是第一个提供稳健理论证明的稳健变量选择方法，但不能简单推广到超高维数据中。因此，本项目试图回答两个问题：1）是否可以构造不依赖高稳健的初始估计的高稳健又高有效的回归估计？ 从而提出针对超高维数据的高稳健又高有效的变量选择方法。进一步考虑它们的多元推广。2）是否能够在非渐进理论的框架下考虑新的稳健性度量，使其能够阐明Quantile回归估计等常用的M-估计的"稳健性"

It is a challenging work to construct a regression estimate with high robustness and high efficiency simultaneously, especilly for Ultra-high dimensional data. Although most M-estimator used in regression models or for (Ultra-) high dimensional data could be claimed to be robust via simulation studies, but their finite sample breakdown point, as an important measure of robustness, is very low, asympotic 0. The estimators based on the step-by-step M-estimators such as MM-estimator and ARETE can overcome this shortcoming to have high robustness and high efficiency simultaneously, but they all rely on an initial estimator with high robustness. The ARETE is the first variable selection method to be verified its robutness in theory, yet it could not be simiply applied in Ultra-high dimensional setting due to the high robustness of initial estimator. So this research try to answer the following two questions:1） Whether can the estimators be provided without the initial estimator with high robustness? And then such estimator can be applied in Ultra-high dimensional setting, also extended to multivariate regression model.2) whether new measure of robustness could be proposed in the framework of non-asysmtotic theory,make it possible to explain the "robustness" of many common used M-estimators such as quantile regression estimators.