  This paper introduces a new regularized version of the robust
$\tau$-regression estimator for analyzing high-dimensional datasets subject to
gross contamination in the response variables and covariates (explanatory
variables). The resulting estimator, termed adaptive $\tau$-Lasso, is robust to
outliers and high-leverage points. It also incorporates an adaptive
$\ell_1$-norm penalty term, which enables the selection of relevant variables
and reduces 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, ensuring both variable-selection
consistency and asymptotic normality. Asymptotic normality applies only to the
entries of the regression vector corresponding to the true support, assuming
knowledge of the true regression vector support. We characterize its robustness
via the finite-sample breakdown point and the influence function. We carry out
extensive simulations and observe that the class of $\tau$-Lasso estimators
exhibits robustness and reliable performance in both contaminated and
uncontaminated data settings. We also validate our theoretical findings on
robustness properties through simulation experiments. In the face of outliers
and high-leverage points, the adaptive $\tau$-Lasso and $\tau$-Lasso estimators
achieve the best performance or close-to-best performance in terms of
prediction and variable selection accuracy compared to other competing
regularized estimators for all scenarios considered in this study. Therefore,
the adaptive $\tau$-Lasso and $\tau$-Lasso estimators can be effectively
employed for a variety of sparse linear regression problems, particularly in
high-dimensional settings and when the data is contaminated by outliers and
high-leverage points.
