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**Aspects of robust linear regression.**
*(English)*
Zbl 0797.62026

This extensive paper on robustness in linear regression is divided into 8 sections. Section 1 contains some general and also critical remarks on robustness and optimality. In Section 2, the influence function of the middle of the shortest half and the Hampel-Rousseeuw least median of squares estimators are derived and it is shown that the existence and boundedness of the influence function will not guarantee stability. In Section 3, linearly invariant weak metrics are constructed and used to calculate breakdown points. The metrics are defined in terms of Vapnik- Cervonenkis classes.

In Section 4, properties of \(S\)-estimators such as existence, bounded solutions, uniqueness and continuity are presented and the breakdown points of these estimators are calculated. It is shown that \(S\)- estimators satisfy an exact Hölder condition of order 1/2 at normal models. In Section 5 the breakdown points of the Hampel-Krasker dispersion and regession functionals are shown to be 0. The exact breakdown point of the Krasker-Welch dispersion functional is calculated (it is not 0) as well as bounds for the corresponding regression functionals.

Section 6 deals with the construction of a globally defined dispersion functional which has the desirable properties to be linearly equivariant, to satisfy a local Lipschitz condition and to have a high breakdown point. In Section 7 the relationship between efficiency, breakdown point and Lipschitz condition is discussed. It is shown that there is no inherent contradiction between efficiency and a high breakdown point, but that there is a conflict between Lipschitz continuity and efficiency. In Section 8 a regression functional is constructed which has a high breakdown point, is linearly equivariant and Lipschitz continuous at models with normal errors, analogously to the dispersion functional in Section 6.

In Section 4, properties of \(S\)-estimators such as existence, bounded solutions, uniqueness and continuity are presented and the breakdown points of these estimators are calculated. It is shown that \(S\)- estimators satisfy an exact Hölder condition of order 1/2 at normal models. In Section 5 the breakdown points of the Hampel-Krasker dispersion and regession functionals are shown to be 0. The exact breakdown point of the Krasker-Welch dispersion functional is calculated (it is not 0) as well as bounds for the corresponding regression functionals.

Section 6 deals with the construction of a globally defined dispersion functional which has the desirable properties to be linearly equivariant, to satisfy a local Lipschitz condition and to have a high breakdown point. In Section 7 the relationship between efficiency, breakdown point and Lipschitz condition is discussed. It is shown that there is no inherent contradiction between efficiency and a high breakdown point, but that there is a conflict between Lipschitz continuity and efficiency. In Section 8 a regression functional is constructed which has a high breakdown point, is linearly equivariant and Lipschitz continuous at models with normal errors, analogously to the dispersion functional in Section 6.

Reviewer: H.Büning (Berlin)

### MSC:

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

62J05 | Linear regression; mixed models |