  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 $\tau$-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.
