Aspects of robust linear regression
type: Item
instance of: scholarly article
title: Aspects of robust linear regression
zbMATH Open document ID: 0797.62026
published in: The Annals of Statistics
publication date: 1994-10-25T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 62F35
Mathematics Subject Classification ID: 62J05
zbMATH DE Number: 528650
zbMATH Keywords: linear invariance
zbMATH Keywords: equivariance
zbMATH Keywords: \(S\)-estimators
zbMATH Keywords: robustness
zbMATH Keywords: optimality
zbMATH Keywords: influence function
zbMATH Keywords: Hampel-Rousseeuw last median of squares estimators
zbMATH Keywords: existence
zbMATH Keywords: boundedness
zbMATH Keywords: stability
zbMATH Keywords: linearly invariant weak metrics
zbMATH Keywords: breakdown points
zbMATH Keywords: Vapnik-Cervonenkis classes
zbMATH Keywords: bounded solutions
zbMATH Keywords: uniqueness
zbMATH Keywords: continuity
zbMATH Keywords: exact Hölder condition
zbMATH Keywords: normal models
zbMATH Keywords: Hampel- Krasker dispersion
zbMATH Keywords: regession functionals
zbMATH Keywords: Krasker-Welch dispersion functional
zbMATH Keywords: globally defined dispersion functional
zbMATH Keywords: local Lipschitz condition
zbMATH Keywords: efficiency
zbMATH Keywords: Lipschitz continuity
author: Laurie Davies
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176349401
OpenAlex ID: W2165952088
DOI: 10.1214/AOS/1176349401
description: scientific article; zbMATH DE number 528650