Robust Variable Selection and Estimation Via Adaptive Elastic Net S-Estimators for Linear Regression

TL;DR

This paper introduces adaptive PENSE, a robust regularized estimator for high-dimensional linear regression that effectively handles heavy-tailed errors and outliers, ensuring reliable variable selection and estimation without prior scale knowledge.

Contribution

The paper proposes a novel adaptive PENSE estimator that achieves the oracle property under minimal assumptions and demonstrates superior robustness and stability in contaminated data scenarios.

Findings

Outperforms existing robust estimators in contaminated data settings

Maintains stability in variable selection under heavy-tailed errors

Shows competitive performance with classical methods on clean data

Abstract

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.