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**Min-max bias robust regression.**
*(English)*
Zbl 0713.62068

Authors’ summary: This paper considers the problem of minimizing the maximum asymptotic bias of regression estimates over \(\epsilon\)- contamination neighborhoods for the joint distribution of the response and carriers. Two classes of estimates are treated: (i) M-estimates with bounded function \(\rho\) applied to the scaled residuals, using a very general class of scale estimates, and (ii) bounded influence function type generalized M-estimates. Estimates in the first class are obtained as the solution of a minimization problem, while estimates in the second class are specified by an estimating equation.

The first class of M-estimates is sufficiently general to include both Huber Proposal 2 simultaneous estimates of regression coefficients and residuals scale, and Rousseeuw-Yohai S-estimates of regression [P. Rousseeuw and V. Yohai, Robust and nonlinear time series analysis, Proc. Workshop. Heidelberg/Ger. 1983, Lect. Notes Stat., Springer-Verlag 26, 256-272 (1984; Zbl 0567.62027)]. It is shown than an S-estimate based on a jump-function type \(\rho\) solves the min-max bias problem for the class of M-estimates with very general scale. This estimate is obtained by the minimization of the \(\alpha\)-quantile of the squared residuals, where \(\alpha =\alpha (\epsilon)\) depends on the fraction of contamination \(\epsilon\). When \(\epsilon\to 0.5\), \(\alpha\) (\(\epsilon\))\(\to 0.5\) and the min-max estimator approaches the least median of squared residuals estimator introduced by P. J. Rousseeuw [J. Am. Stat. Assoc. 79, 871-880 (1984; Zbl 0547.62046)].

For the bounded influence class of GM-estimates it is shown the “sign” type nonlinearity yields the min-max estimate. This estimate coincides with the minimum gross-error sensitivity GM-estimate. For \(p=1\), the optimal GM-estimate is optimal among the class of all equivariant regression estimates. The min-max S-estimator has a breakdown point which is independent of the number of carriers p and tends to 0.5 as \(\epsilon\) increases to 0.5, but has a slow rate of convergence. The min-max GM- estimate has the usual rate of convergence, but a breakdown point which decreases to zero with increasing p. Finally, we compare the min-max biases for both types of estimates, for the case where the nominal model is multivariate normal.

The first class of M-estimates is sufficiently general to include both Huber Proposal 2 simultaneous estimates of regression coefficients and residuals scale, and Rousseeuw-Yohai S-estimates of regression [P. Rousseeuw and V. Yohai, Robust and nonlinear time series analysis, Proc. Workshop. Heidelberg/Ger. 1983, Lect. Notes Stat., Springer-Verlag 26, 256-272 (1984; Zbl 0567.62027)]. It is shown than an S-estimate based on a jump-function type \(\rho\) solves the min-max bias problem for the class of M-estimates with very general scale. This estimate is obtained by the minimization of the \(\alpha\)-quantile of the squared residuals, where \(\alpha =\alpha (\epsilon)\) depends on the fraction of contamination \(\epsilon\). When \(\epsilon\to 0.5\), \(\alpha\) (\(\epsilon\))\(\to 0.5\) and the min-max estimator approaches the least median of squared residuals estimator introduced by P. J. Rousseeuw [J. Am. Stat. Assoc. 79, 871-880 (1984; Zbl 0547.62046)].

For the bounded influence class of GM-estimates it is shown the “sign” type nonlinearity yields the min-max estimate. This estimate coincides with the minimum gross-error sensitivity GM-estimate. For \(p=1\), the optimal GM-estimate is optimal among the class of all equivariant regression estimates. The min-max S-estimator has a breakdown point which is independent of the number of carriers p and tends to 0.5 as \(\epsilon\) increases to 0.5, but has a slow rate of convergence. The min-max GM- estimate has the usual rate of convergence, but a breakdown point which decreases to zero with increasing p. Finally, we compare the min-max biases for both types of estimates, for the case where the nominal model is multivariate normal.

Reviewer: Guijing Chen

### MSC:

62J02 | General nonlinear regression |

62F35 | Robustness and adaptive procedures (parametric inference) |

62F10 | Point estimation |

62J05 | Linear regression; mixed models |

62F12 | Asymptotic properties of parametric estimators |