We introduce a new class of robust M-estimators for performing simultaneous parameter estimation and variable selection in high-dimensional regression models. We first explain the motivations for the key ingredient of our procedures which are inspired by regularization methods used in wavelet thresholding in noisy signal processing. The derived penalized estimation procedures are shown to enjoy theoretically the oracle property both in the classical finite dimensional case as well as the high-dimensional case when the number of variables p is not fixed but can grow with the sample size n, and to achieve optimal asymptotic rates of convergence. A fast accelerated proximal gradient algorithm, of coordinate descent type, is proposed and implemented for computing the estimates and appears to be surprisingly efficient in solving the corresponding regularization problems including the case for ultra high-dimensional data where \(p \gg n\). Finally, a very extensive simulation study and some real data analysis, compare several recent existing M-estimation procedures with the ones proposed in the paper, and demonstrate their utility and their advantages.

我们引入一类新的鲁棒M估计器，以在高维回归模型中执行同时参数估计和变量选择。我们首先解释程序中关键成分的动机，这些动机是由在噪声信号处理中的小波阈值化中使用的正则化方法所启发的。当变量p的数量不固定但可以随样本大小n的增加而增加时，在经典的有限维情况和高维情况下，导出的惩罚估计程序在理论上都显示出预言性。，以达到最佳的渐近收敛速度。提出并实现了坐标下降型的快速加速近端梯度算法，该算法用于计算估计值，并且在解决相应的正则化问题（包括超高维数据的情况下，其中\（p \ gg n \）的情况下）似乎出奇地有效。。最后，进行了非常广泛的仿真研究和一些实际数据分析，将最近几种现有的M估计程序与本文中提出的程序进行了比较，并证明了它们的实用性和优势。