Breakdown and groups

P. Laurie Davies; Ursula Gather; Marc G. Genton; André Lucas; Frank Hampel; H. E. Xuming; Hannu Oja; Peter J. Rousseeuw; David E. Tyler

doi:10.1214/009053604000001138

Breakdown and groups

P. Laurie Davies^*, Ursula Gather, Marc G. Genton, André Lucas, Frank Hampel, H. E. Xuming, Hannu Oja, Peter J. Rousseeuw, David E. Tyler

^*Corresponding author for this work

Research output: Contribution to journal › Review article › peer-review

73 Scopus citations

Abstract

The concept of breakdown point was introduced by Hampel [Ph.D. dissertation (1968), Univ. California, Berkeley; Ann. Math. Statist. 42 (1971) 1887-1896] and developed further by, among others, Huber [Robust Statistics (1981). Wiley, New York] and Donoho and Huber [In A Festschrift for Erich L. Lehmann (1983) 157-184. Wadsworth, Belmont, CA]. It has proved most successful in the context of location, scale and regression problems. Attempts to extend the concept to other situations have not met with general acceptance. In this paper we argue that this is connected to the fact that in the location, scale and regression problems the translation and affine groups give rise to a definition of equivariance for statistical functionals. Comparisons in terms of breakdown points seem only useful when restricted to equivariant functionals and even here the connection between breakdown and equivariance is a tenuous one.

Original language	English (US)
Pages (from-to)	977-1035
Number of pages	59
Journal	Annals of Statistics
Volume	33
Issue number	3
DOIs	https://doi.org/10.1214/009053604000001138
State	Published - Jun 2005
Externally published	Yes

Keywords

Breakdown point
Equivariance
Robust statistics

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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10.1214/009053604000001138

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title = "Breakdown and groups",

abstract = "The concept of breakdown point was introduced by Hampel [Ph.D. dissertation (1968), Univ. California, Berkeley; Ann. Math. Statist. 42 (1971) 1887-1896] and developed further by, among others, Huber [Robust Statistics (1981). Wiley, New York] and Donoho and Huber [In A Festschrift for Erich L. Lehmann (1983) 157-184. Wadsworth, Belmont, CA]. It has proved most successful in the context of location, scale and regression problems. Attempts to extend the concept to other situations have not met with general acceptance. In this paper we argue that this is connected to the fact that in the location, scale and regression problems the translation and affine groups give rise to a definition of equivariance for statistical functionals. Comparisons in terms of breakdown points seem only useful when restricted to equivariant functionals and even here the connection between breakdown and equivariance is a tenuous one.",

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AU - Davies, P. Laurie

AU - Gather, Ursula

AU - Genton, Marc G.

AU - Lucas, André

AU - Hampel, Frank

AU - Xuming, H. E.

AU - Oja, Hannu

AU - Rousseeuw, Peter J.

AU - Tyler, David E.

PY - 2005/6

Y1 - 2005/6

N2 - The concept of breakdown point was introduced by Hampel [Ph.D. dissertation (1968), Univ. California, Berkeley; Ann. Math. Statist. 42 (1971) 1887-1896] and developed further by, among others, Huber [Robust Statistics (1981). Wiley, New York] and Donoho and Huber [In A Festschrift for Erich L. Lehmann (1983) 157-184. Wadsworth, Belmont, CA]. It has proved most successful in the context of location, scale and regression problems. Attempts to extend the concept to other situations have not met with general acceptance. In this paper we argue that this is connected to the fact that in the location, scale and regression problems the translation and affine groups give rise to a definition of equivariance for statistical functionals. Comparisons in terms of breakdown points seem only useful when restricted to equivariant functionals and even here the connection between breakdown and equivariance is a tenuous one.

AB - The concept of breakdown point was introduced by Hampel [Ph.D. dissertation (1968), Univ. California, Berkeley; Ann. Math. Statist. 42 (1971) 1887-1896] and developed further by, among others, Huber [Robust Statistics (1981). Wiley, New York] and Donoho and Huber [In A Festschrift for Erich L. Lehmann (1983) 157-184. Wadsworth, Belmont, CA]. It has proved most successful in the context of location, scale and regression problems. Attempts to extend the concept to other situations have not met with general acceptance. In this paper we argue that this is connected to the fact that in the location, scale and regression problems the translation and affine groups give rise to a definition of equivariance for statistical functionals. Comparisons in terms of breakdown points seem only useful when restricted to equivariant functionals and even here the connection between breakdown and equivariance is a tenuous one.

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KW - Equivariance

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JF - Annals of Statistics

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ER -

Breakdown and groups

Abstract

Keywords

ASJC Scopus subject areas

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