A practical method of robust estimation in case of asymmetry

Abstract

A modified Huber loss function is introduced in order to make robust location and regression estimation in the case of asymmetry in a possibly contaminated distribution. This loss function involves a measurement of asymmetry r, which the unequal weights of the observed information can depend upon. Based on this new function, the M-estimators derived can enhance robustness against possible departures from the model assumption and outliers. In order to accommodate asymmetry, a practical method for estimating r is presented and an iterated procedure for both location and regression estimation is developed. Simulation studies are executed on the effects of the proposed modification in the Huber loss function for robust estimation of location parameter for a possibly contaminated distribution and the coefficient parameters in linear regression with errors from a possibly contaminated distribution. We use bias, variance, mean squared error, and relative efficiency as criteria in our simulation study in order to compare the newly developed estimators with the original Huber’s robust estimators. A set of suitable tuning constants used for this modified Huber function has been attained. The results have shown that the proposed M-estimators are efficient. An application to a real data set is also presented in order to illustrate the proposed methods.