A review on robust M-estimators for regression analysis

Abstract Regression analysis constitutes an important tool for investigating the effect of explanatory variables on response variables. When outliers and bias errors are present, the weighted least squares estimator can perform poorly. For this reason, alternative robust techniques have been studied in several areas of science. However, often these different scientific communities are disconnected from each other, culminating in the scarcity of knowledge exchange among these areas. Thus, this paper presents a review on robust M-estimators in various knowledge areas. 50 (48 robust) M-estimators are illustrated, including the Weighted Least Squares estimator (non-robust), the Contaminated Normal estimator (quasi-robust), the Huber estimator (monotone), the Correntropy estimator (soft-redescending), the Smith estimator (hard-redescending), and the adaptive Barron and Generalized T-distribution. The mathematical functions that describe the estimators and their respective graphical forms are presented. The tuning constants of all these estimators, for 90%, 95%, 98%, and 99% relative efficiency levels in respect to the Normal distribution are also presented.