This paper introduces a new regularized version of the robust $\tau$-regression estimator for analyzing high-dimensional data sets subject to gross contamination in the response variables and covariates. We call the resulting estimator adaptive $\tau$-Lasso that is robust to outliers and high-leverage points and simultaneously employs adaptive $\ell_1$-norm penalty term to reduce the bias associated with large true regression coefficients. More specifically, this adaptive $\ell_1$-norm penalty term assigns a weight to each regression coefficient. For a fixed number of predictors $p$, we show that the adaptive $\tau$-Lasso has the oracle property with respect to variable-selection consistency and asymptotic normality for the regression vector corresponding to the true support, assuming knowledge of the true regression vector support. We then characterize its robustness via the finite-sample breakdown point and the influence function. We carry-out extensive simulations to compare the performance of the adaptive $\tau$-Lasso estimator with that of other competing regularized estimators in terms of prediction and variable selection accuracy in the presence of contamination within the response vector/regression matrix and additive heavy-tailed noise. We observe from our simulations that the class of $\tau$-Lasso estimators exhibits robustness and reliable performance in both contaminated and uncontaminated data settings, achieving the best or close-to-best for many scenarios, except for oracle estimators. However, it is worth noting that no particular estimator uniformly dominates others. We also validate our findings on robustness properties through simulation experiments.

本文提出了一个新的正则化的自适应tau-Lasso估计器，用于分析响应变量和协变量受到强烈污染的高维数据集，能够同时使用自适应L1范数惩罚项降低大真实回归系数与降低偏差的关联性，具有良好的鲁棒性和可靠性，表现比同类估计器更好或接近最佳，特别是在污染数据和期望回归矩阵/响应向量的情况下。同时也在探索鲁棒性质方面进行了验证。

自适应$τ$-Lasso: 其鲁棒性和Oracle特性