The Annals of Statistics

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

José R. Berrendero, Beatriz V. M. Mendes, and David E. Tyler

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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.

Article information

Ann. Statist., Volume 35, Number 1 (2007), 13-40.

First available in Project Euclid: 6 June 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F35: Robustness and adaptive procedures
Secondary: 62J05: Linear regression

Robust regression M-estimates S-estimates constrained M-estimates maximum bias curves breakdown point gross error sensitivity


Berrendero, José R.; Mendes, Beatriz V. M.; Tyler, David E. On the maximum bias functions of MM -estimates and constrained M -estimates of regression. Ann. Statist. 35 (2007), no. 1, 13--40. doi:10.1214/009053606000000975.

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