Robust regression with asymmetric loss functions

In robust regression, it is usually assumed that the distribution of the error term is symmetric or the data are symmetrically contaminated by outliers. However, this assumption is usually not satisfied in practical problems, and thus if the traditional robust methods, such as Tukey's biweight and Huber's method, are used to estimate the regression parameters, the efficiency of the parameter estimation can be lost. In this paper, we construct an asymmetric Tukey's biweight loss function with two tuning parameters and propose a data-driven method to find the most appropriate tuning parameters. Furthermore, we provide an adaptive algorithm to obtain robust and efficient parameter estimates. Our extensive simulation studies suggest that the proposed method performs better than the symmetric methods when error terms follow an asymmetric distribution or are asymmetrically contaminated. Finally, a cardiovascular risk factors dataset is analyzed to illustrate the proposed method.