Weighted Least Absolute Deviation Regression Estimator with the SCAD Function

In a regression model the estimator based on the absolute deviation loss function is more robust than that based on the squares error loss function. However, the least absolute deviation estimator is sensitive to leverage points of the predictors, even though it is robust to the regression outliers. We propose a robust penalized regression estimator to regression outliers and leverage points which provides automatically selection of variables together. It is based on the weighted least absolute deviation (SCAD) and the non-convex penalty function, the smoothly clipped absolute deviation function which has the oracle property. We develop a unified algorithm for the proposed estimator including the SCAD estimate, based on the local quadratic approximation and the tuning parameter of the penalty function. Numerical simulation shows that the proposed estimator is effective for analysing a contaminated data.