  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.
