Robust Variable Selection and Estimation Via Adaptive Elastic Net S-Estimators for Linear Regression
This work addresses the need for reliable statistical analysis in high-dimensional regression under adverse conditions like heavy-tailed errors and outliers, offering an incremental improvement over existing robust methods.
The authors tackled the problem of robust variable selection and estimation in high-dimensional linear regression with heavy-tailed errors and outliers by proposing the adaptive PENSE estimator, which demonstrated superior performance in contaminated samples and remained competitive in clean samples.
Heavy-tailed error distributions and predictors with anomalous values are ubiquitous in high-dimensional regression problems and can seriously jeopardize the validity of statistical analyses if not properly addressed. For more reliable estimation under these adverse conditions, we propose a new robust regularized estimator for simultaneous variable selection and coefficient estimation. This estimator, called adaptive PENSE, possesses the oracle property without prior knowledge of the scale of the residuals and without any moment conditions on the error distribution. The proposed estimator gives reliable results even under very heavy-tailed error distributions and aberrant contamination in the predictors or residuals. Importantly, even in these challenging settings variable selection by adaptive PENSE remains stable. Numerical studies on simulated and real data sets highlight superior finite-sample performance in a vast range of settings compared to other robust regularized estimators in the case of contaminated samples and competitiveness compared to classical regularized estimators in clean samples.