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High-Breakdown Estimators of Multivariate Location and Scatter

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Robustness and Complex Data Structures

Abstract

This contribution gives a brief summary of robust estimators of multivariate location and scatter. We assume that the original (uncontaminated) data follow an elliptical distribution with location vector μ and positive definite scatter matrix Σ. Robust methods aim to estimate μ and Σ even though the data has been contaminated by outliers. The well-known multivariate M-estimators can break down when the outlier fraction exceeds 1/(p+1) where p is the number of variables. We describe several robust estimators that can withstand a high fraction (up to 50 %) of outliers, such as the minimum covariance determinant estimator (MCD), the Stahel–Donoho estimator, S-estimators and MM-estimators. We also discuss faster methods that are only approximately equivariant under linear transformations, such as the orthogonalized Gnanadesikan–Kettenring estimator and the deterministic MCD algorithm.

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Authors and Affiliations

  1. Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001, Heverlee, Belgium

    Peter Rousseeuw & Mia Hubert

Authors
  1. Peter Rousseeuw
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  2. Mia Hubert
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Correspondence to Peter Rousseeuw .

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Editors and Affiliations

  1. Halle-Wittenberg, Faculty of Law and Economics, Martin-Luther-University, Große Steinstraße 73, Halle, 06099, Germany

    Claudia Becker

  2. Faculty of Statistics, TU Dortmund University, Vogelpothsweg 87, Dortmund, 44227, Germany

    Roland Fried

  3. Faculty of Statistics, TU Dortmund University, Vogelpothsweg 87, Dortmund, 44227, Germany

    Sonja Kuhnt

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Rousseeuw, P., Hubert, M. (2013). High-Breakdown Estimators of Multivariate Location and Scatter. In: Becker, C., Fried, R., Kuhnt, S. (eds) Robustness and Complex Data Structures. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35494-6_4

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