Robust approach is often desirable in presence of outliers for more efficient parameter estimation. However, the choice of the regularization parameter value impacts the efficiency of the parameter estimators. To maximize the estimation efficiency, we construct a likelihood function for simultaneously estimating the regression parameters and the tuning parameter. The "working" likelihood function is deemed as a vehicle for efficient regression parameter estimation, because we do not assume the data are generated from this likelihood function. The proposed method can effectively find a value of the regularization parameter based on the extent of contamination in the data. We carry out extensive simulation studies in a variety of cases to investigate the performance of the proposed method. The simulation results show that the efficiency can be enhanced as much as 40% when the data follow a heavy-tailed distribution, and reaches as high as 468% for the heteroscedastic variance cases compared to the traditional Huber's method with a fixed regularization parameter. For illustration, we also analyzed two datasets: one from a diabetics study and the other from a mortality study.
Keywords: Data driven; Huber’s loss function; robust method; tuning parameter; working likelihood.