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Tyler Shape Depth

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  • Davy Paindaveine
  • Germain Van Bever

Abstract

In many problems from multivariate analysis (principal component analysis, testing for sphericity, etc.), the parameter of interest is a shape matrix, that is, a normalised version of the corresponding scatter or dispersion matrix. In this paper, we propose a depth concept for shape matrices which is of a sign nature, in the sense that it involves data points only through their directions from the center of the distribution. We use the terminology Tyler shape depth since the resulting estimator of shape — namely, the deepest shape matrix — is the depth-based counterpart of the celebrated M-estimator of shape from Tyler (1987). We in- vestigate the invariance, quasi-concavity and continuity properties of Tyler shape depth, as well as the topological and boundedness properties of the corresponding depth regions. We study existence of a deepest shape matrix and prove Fisher consistency in the elliptical case. We derive a Glivenko-Cantelli-type result and establish the almost sure consistency of the deepest shape matrix estimator. We also consider depth-based tests for shape and investigate their finite-sample per- formances through simulations. Finally, we illustrate the practical relevance of the proposed depth concept on a real data example.

Suggested Citation

  • Davy Paindaveine & Germain Van Bever, 2017. "Tyler Shape Depth," Working Papers ECARES ECARES 2017-29, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2013/255000
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    References listed on IDEAS

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    1. Thomas P. Hettmansperger, 2002. "A practical affine equivariant multivariate median," Biometrika, Biometrika Trust, vol. 89(4), pages 851-860, December.
    2. 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.
    3. 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.
    4. 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.
    5. Paindaveine, Davy, 2008. "A canonical definition of shape," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2240-2247, October.
    6. Zhang, Jian, 2002. "Some Extensions of Tukey's Depth Function," Journal of Multivariate Analysis, Elsevier, vol. 82(1), pages 134-165, July.
    7. Lutz Dümbgen, 1998. "On Tyler's M-Functional of Scatter in High Dimension," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 50(3), pages 471-491, September.
    8. Marc Hallin & Davy Paindaveine & Miroslav Siman, 2008. "Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth," Working Papers ECARES 2008_042, ULB -- Universite Libre de Bruxelles.
    9. Müller, Christine H., 2005. "Depth estimators and tests based on the likelihood principle with application to regression," Journal of Multivariate Analysis, Elsevier, vol. 95(1), pages 153-181, July.
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    1. repec:eee:stapro:v:129:y:2017:i:c:p:335-339 is not listed on IDEAS

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    Keywords

    Elliptical distribution; Robustness; Shape matrix; Statistical depth; Test for sphericity;

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