Penalized MM Regression Estimation with $L_{\gamma }$ Penalty: A Robust Version of Bridge Regression

The bridge regression estimator generalizes both ridge regression and LASSO estimators. Since it minimizes the sum of squared residuals with a $L_{\gamma }$ penalty, this estimator is typically not robust against outliers in the data. There have been attempts to define robust versions of the bridge regression method, but while these proposed methods produce bridge regression estimators robust to outliers and heavy-tailed errors, they are not robust against leverage points. We propose a robust bridge regression estimation method combining MM and bridge regression estimation methods. The MM bridge regression estimator obtained from the proposed method is robust against outliers and leverage points. Furthermore, for appropriate choices of the penalty function, the proposed method is able to perform variable selection and parameter estimation simultaneously. Consistency, asymptotic normality, and sparsity of the MM bridge regression estimator are achieved. We propose an algorithm to compute the MM bridge regression estimate. A simulation study and a real data example are provided to demonstrate the performance of the MM bridge regression estimator for finite sample cases.