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Thek-step spatial sign covariance matrix

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Abstract

The Sign Covariance Matrix is an orthogonal equivariant estimator of multivariate scale. It is often used as an easy-to-compute and highly robust estimator. In this paper we propose ak-step version of the Sign Covariance Matrix, which improves its efficiency while keeping the maximal breakdown point. Ifk tends to infinity, Tyler’s M-estimator is obtained. It turns out that even for very low values ofk, one gets almost the same efficiency as Tyler’s M-estimator.

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Article08 September 2023
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References

  • Bensmail H, Celeux G (1996) Regularized Gaussian discriminant analysis through eigenvalue decomposition. J Am Stat Assoc 91: 1743–1749

    Article MATH MathSciNet  Google Scholar 

  • Croux C, Haesbroeck G (2000) Principal component analysis based on robust estimators of the covariance or correlation matrix: influence functions and efficiencies. Biometrika 87: 603–618

    Article MATH MathSciNet  Google Scholar 

  • Croux C, Ollila E, Oja H (2002) Sign and rank covariance matrices: statistical properties and application to principal components analysis. In: Dodge Y (eds) Statistical data analysis based on the L1-norm and related methods. Birkhauser, Basel, pp 257–271

    Google Scholar 

  • Dümbgen L, Tyler DE (2005) On the breakdown properties of some multivariate M-functionals. Scand J Stat 32: 247–264

    Article MATH  Google Scholar 

  • Frahm G (2009) Asymptotic distiributions of robust shape matrices and scale. J Multivar Anal 100: 1329–1337

    Article MATH MathSciNet  Google Scholar 

  • Hallin M, Oja H, Paindaveine D (2006) Semiparametrically efficient rank-based inference for shape. II. Optimal R-estimation of shape. Ann Stat 34: 2757–2789

    Article MATH MathSciNet  Google Scholar 

  • Hettmansperger TP, Randles RH (2002) A practical affine equivariant multivariate médian. Biometrika 89(4): 851–860

    Article MATH MathSciNet  Google Scholar 

  • Kent JT, Tyler DE (1991) Redescending M-estimates of multivariate location and scatter. Ann Stat 19: 2102–2119

    Article MATH MathSciNet  Google Scholar 

  • Locantore N, Marron JS, Simpson DG, Tripoli N, Zhang JT, Cohen KL (1999) Robust principal components for functional data. Test 8: 1–28

    Article MATH MathSciNet  Google Scholar 

  • Lopuhäa HP, Rousseeuw PJ (1991) Breakdown points of affine equivariant estimators of multivariate location and covariance matrices. Ann Stat 19: 229–248

    Article  Google Scholar 

  • Maronna R, Martin D, Yohai V (2006) Robust statistics. Wiley, New York

    Book MATH  Google Scholar 

  • Oja H (2010) Multivariate nonparametric methods with R. An approach based on spatial signs and ranks, Springer, Berlin (in press)

  • Ollila E, Heetmansperger TP, Oja H (2002) Affine equivariant multivariate sign methods (unpublished)

  • Paindaveine D (2008) A canonical definition of shape. Stat Probab Lett 78: 2240–2247

    Article MATH MathSciNet  Google Scholar 

  • Rousseeuw PJ, Croux C (1994) The bias of k-step M-estimators. Stat Probab Lett 20: 411–420

    Article MATH MathSciNet  Google Scholar 

  • Sirkia S, Taskinen S, Oja H, Tyler D (2009) Tests and estimates of shape based on spatial signs and ranks. J Nonparametr Stat 21: 15–176

    Article MathSciNet  Google Scholar 

  • Taskinen S, Sirkiä S, Oja H (2010) k-step Shape estimators based on spatial signs and ranks. J Stat Plan Inference (in press)

  • Tyler DE (1987) A distribution-free M-estimator of multivariate scatter. Ann Stat 15: 234–251

    Article MATH MathSciNet  Google Scholar 

  • Visuri S, Ollila E, Koivunen V, Möttönen J, Oja H (2003) Affine equivariant multivariate rank methods. J Stat Plan Infer 114: 161–185

    Article MATH  Google Scholar 

  • Yadine A (2006) Robustness and efficiency of multivariate scatter estimators”, Ph.D. dissertation, Université Libre de Bruxelles, Bruxelles

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This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Authors and Affiliations

  1. Faculty of Business and Economics, K.U. Leuven, Louvain, Belgium

    C. Croux

  2. Faculty of Economics and Business Administration, Tilburg University, Tilburg, The Netherlands

    C. Croux

  3. Institut de Recherche en Statistique, ECARES, Université libre de Bruxelles, CP-114, Av. F.D. Roosevelt 50, 1050, Brussels, Belgium

    C. Dehon

  4. Institut de Recherche en Statistique, Université libre de Bruxelles, Av. F.D. Roosevelt 50, 1050, Brussels, Belgium

    A. Yadine

Authors
  1. C. Croux

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  2. C. Dehon

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  3. A. Yadine

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Corresponding author

Correspondence toC. Croux.

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Croux, C., Dehon, C. & Yadine, A. Thek-step spatial sign covariance matrix.Adv Data Anal Classif4, 137–150 (2010). https://doi.org/10.1007/s11634-010-0062-7

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