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On the global robustness of generalized S-estimators. (English) Zbl 1004.62025

Summary: A generalized \(S\)-estimator \((GS\)-estimator) of regression is obtained by minimizing an \(M\)-estimator of scale applied to the pairwise differences of residuals \(r_i(\theta)-r_j (\theta)\), \(i<j\). We focus on the global robustness properties of these estimators. It was pointed out by C. Croux et al. [J. Am. Stat. Assoc. 89, No. 428, 1271-1281 (1994; Zbl 0812.62073)] that two \(GS\)-estimators with the same breakdown point may have very different global robustness behavior. Accordingly, we supplement the information given by the breakdown point with the explosion rate, a summary measure of the robustness behavior of the estimator when the amount of contamination is large.
We provide formulas that allow the computation of explosion rates and establish a link between the local behavior at zero of the score function on which the \(M\)-estimator is based and the robustness of the corresponding \(GS\)-estimator. Finally, we apply the explosion rate to quantify the loss of robustness when using \(GS\)-estimators instead of the simpler non-generalized \(S\)-estimators.

MSC:

62F35 Robustness and adaptive procedures (parametric inference)
62J05 Linear regression; mixed models
62F10 Point estimation

Citations:

Zbl 0812.62073
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References:

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