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Breakdown and groups

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  • Davies, P. Laurie
  • Gather, U.

Abstract

The concept of breakdown point was introduced by Hodges (1967) and Hampel (1968, 1971) and still plays an important though at times a controversial role in robust statistics. It has proved most successful in the context of location, scale and regression problems. In this paper we argue that this success is intimately connected to the fact that the translation and affine groups act on the sample space and give rise to a definition of equivariance for statistical functionals. For such functionals a nontrivial upper bound for the breakdown point can be shown. In the absence of such a group structure a breakdown point of one is attainable and this is perhaps the decisive reason why the concept of breakdown point in other situations has not proved as successful. Even if a natural group is present it is often not sufficiently large to allow a nontrivial upper bound for the breakdown point. One exception to this is the problem of the autocorrelation structure of time series where we derive a nontrivial upper breakdown point using the group of realizable linear filters. The paper is formulated in an abstract manner to emphasize the role of the group and the resulting equivariance structure.

Suggested Citation

  • Davies, P. Laurie & Gather, U., 2002. "Breakdown and groups," Technical Reports 2002,57, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    • Handle: RePEc:zbw:sfb475:200257
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    References listed on IDEAS

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    1. Davies, P. Laurie & Kovac, A., 2002. "Densities, spectral densities and modality," Technical Reports 2002,53, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    2. Hubert, Mia, 1997. "The breakdown value of the L1 estimator in contingency tables," Statistics & Probability Letters, Elsevier, vol. 33(4), pages 419-425, May.
    3. Yanyuan Ma & Marc G. Genton, 2000. "Highly Robust Estimation of the Autocovariance Function," Journal of Time Series Analysis, Wiley Blackwell, vol. 21(6), pages 663-684, November.
    4. He, Xuming & Fung, Wing K., 2000. "High Breakdown Estimation for Multiple Populations with Applications to Discriminant Analysis," Journal of Multivariate Analysis, Elsevier, vol. 72(2), pages 151-162, February.
    5. Gordaliza, A., 1991. "On the breakdown point of multivariate location estimators based on trimming procedures," Statistics & Probability Letters, Elsevier, vol. 11(5), pages 387-394, May.
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    1. Davies, P. Laurie, 2002. "Statistical procedures and robust statistics," Technical Reports 2002,54, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.

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