On the Maximum Bias Functions of Mm-estimates and Constrained M-estimates of Regression

We derive the maximum bias functions of the MM -estimates and the constrained M -estimates or CM -estimates of regression and compare them to the maximum bias functions of the S -estimates and the τ -estimates of regression. In these comparisons, the CM -estimates tend to exhibit the most favorable bias-robustness properties. Also, under the Gaussian model, it is shown how one can construct a CM -estimate which has a smaller maximum bias function than a given S -estimate, that is, the resulting CM -estimate dominates the S -estimate in terms of maxbias and, at the same time, is considerably more efficient.