On the maximum bias functions of MM-estimates and constrained M-estimates of regression

TL;DR

This paper derives and compares the maximum bias functions of various robust regression estimators, showing that CM-estimates often outperform others in bias-robustness and efficiency under Gaussian models.

Contribution

It provides a detailed derivation of maximum bias functions for MM, CM, S, and τ-estimates, highlighting the superior robustness and efficiency of CM-estimates.

Findings

CM-estimates have the most favorable bias-robustness properties.

It is possible to construct CM-estimates with smaller maximum bias than S-estimates.

CM-estimates can be more efficient while maintaining robustness.

Abstract

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.