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On the Breakdown Properties of Some Multivariate M‐Functionals

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  • LUTZ DÜMBGEN
  • DAVID E. TYLER

Abstract

. For probability distributions on ℝq, a detailed study of the breakdown properties of some multivariate M‐functionals related to Tyler's [Ann. Statist. 15 (1987) 234] ‘distribution‐free’ M‐functional of scatter is given. These include a symmetrized version of Tyler's M‐functional of scatter, and the multivariate t M‐functionals of location and scatter. It is shown that for ‘smooth’ distributions, the (contamination) breakdown point of Tyler's M‐functional of scatter and of its symmetrized version are 1/q and , respectively. For the multivariate t M‐functional which arises from the maximum likelihood estimate for the parameters of an elliptical t distribution on ν ≥ 1 degrees of freedom the breakdown point at smooth distributions is 1/(q + ν). Breakdown points are also obtained for general distributions, including empirical distributions. Finally, the sources of breakdown are investigated. It turns out that breakdown can only be caused by contaminating distributions that are concentrated near low‐dimensional subspaces.

Suggested Citation

  • Lutz Dümbgen & David E. Tyler, 2005. "On the Breakdown Properties of Some Multivariate M‐Functionals," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(2), pages 247-264, June.
    • Handle: RePEc:bla:scjsta:v:32:y:2005:i:2:p:247-264
    DOI: 10.1111/j.1467-9469.2005.00425.x
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    Cited by:

    1. Davy Paindaveine & Germain Van Bever, 2013. "Inference on the Shape of Elliptical Distribution Based on the MCD," Working Papers ECARES ECARES 2013-27, ULB -- Universite Libre de Bruxelles.
    2. Roelant, E. & Van Aelst, S. & Croux, C., 2009. "Multivariate generalized S-estimators," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 876-887, May.
    3. Robert Serfling & Satyaki Mazumder, 2013. "Computationally easy outlier detection via projection pursuit with finitely many directions," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 25(2), pages 447-461, June.
    4. Frahm, Gabriel, 2009. "Asymptotic distributions of robust shape matrices and scales," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1329-1337, August.
    5. Paindaveine, Davy, 2009. "On Multivariate Runs Tests for Randomness," Journal of the American Statistical Association, American Statistical Association, vol. 104(488), pages 1525-1538.
    6. C. Croux & C. Dehon & A. Yadine, 2010. "The k-step spatial sign covariance matrix," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 4(2), pages 137-150, September.
    7. Frahm, Gabriel & Jaekel, Uwe, 2010. "A generalization of Tyler's M-estimators to the case of incomplete data," Computational Statistics & Data Analysis, Elsevier, vol. 54(2), pages 374-393, February.
    8. Davy Paindaveine & Germain Van Bever, 2017. "Halfspace Depths for Scatter, Concentration and Shape Matrices," Working Papers ECARES ECARES 2017-19, ULB -- Universite Libre de Bruxelles.
    9. Hallin, Marc & Paindaveine, Davy, 2009. "Optimal tests for homogeneity of covariance, scale, and shape," Journal of Multivariate Analysis, Elsevier, vol. 100(3), pages 422-444, March.
    10. Seija Sirkiä & Sara Taskinen & Hannu Oja & David Tyler, 2009. "Tests and estimates of shape based on spatial signs and ranks," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(2), pages 155-176.
    11. Hallin Marc & Paindaveine Davy, 2006. "Parametric and semiparametric inference for shape: the role of the scale functional," Statistics & Risk Modeling, De Gruyter, vol. 24(3), pages 1-24, December.
    12. Paindaveine, Davy & Van Bever, Germain, 2014. "Inference on the shape of elliptical distributions based on the MCD," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 125-144.
    13. Taskinen, Sara & Koch, Inge & Oja, Hannu, 2012. "Robustifying principal component analysis with spatial sign vectors," Statistics & Probability Letters, Elsevier, vol. 82(4), pages 765-774.
    14. Sirkiä, Seija & Taskinen, Sara & Oja, Hannu, 2007. "Symmetrised M-estimators of multivariate scatter," Journal of Multivariate Analysis, Elsevier, vol. 98(8), pages 1611-1629, September.
    15. Paindaveine, Davy, 2008. "A canonical definition of shape," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2240-2247, October.

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