Constrained M-Estimation for Regression

When using redescending M-estimates of regression, one must choose not only an estimate of scale, but since the redescending M-estimating equations may admit multiple solutions, of which all of them may not be a desired solution, one must also have a method for choosing a desirable solution to the estimating equations. We introduce here a new approach for properly scaling redescending M-estimating equations and for obtaining high breakdown point solutions to the equations by the introduction of the constrained M-estimates of regression, or the CM-estimates of regression for short. Unlike the S-estimates of regression, the CM-estimates of regression can be tuned to obtain good local robustness properties while maintaining a breakdown point of 1/2.