A robust adaptive Lasso estimator for the independent contamination model

Abstract

The Lasso has become a benchmark method for simultaneous parameter estimation and variable selection in regression analysis. It is based on the least-squares estimator and, therefore, suffers from the presence of outliers. Robust Lasso methods combine the objective function of a robust estimator with ℓ1-penalization. We address robustness for cases in which the number of observations is smaller (or not much larger) than the number of predictors. Further, we assume that the regression matrix may contain cellwise outliers. In such settings, even a few highly contaminated predictors can cause existing robust methods that are based on the commonly used rowwise contamination model to break down. Therefore, we propose a new adaptive Lasso type regularization. It takes into account cellwise outlyingness in the regression matrix and uses this information for robust variable selection. The proposed regularization term is integrated into the objective function of the MM-estimator, which yields the proposed MM-Robust Weighted Adaptive Lasso (MM-RWAL). A performance comparison to existing robust Lasso estimators is provided using Monte Carlo experiments. Further, the MM-RWAL is applied to determine the temporal releases of the European Tracer Experiment (ETEX) at the source location. This sparse and ill-conditioned linear inverse problem contains cellwise and rowwise outliers.