M-estimators are robust estimators that give less weight to the observations that are outliers while redescending M-estimators are those estimators that are built such that extreme outliers are completely rejected. In this paper, redescending M-estimators are compared using both the Monte Carlo simulation method and the real life data to ascertain the method that is more efficient and robust when outliers are in both x and y directions. The results from the simulation study and the real life data indicate that Anekwe redescending M-estimator is more efficient and robust when outliers are in both x and y directions.