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A canonical definition of shape

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  • Paindaveine, Davy

Abstract

Very general concepts of scatter, extending the traditional notion of covariance matrices, have become classical tools in robust multivariate analysis. In many problems of practical importance (principal components, canonical correlation, testing for sphericity), only homogeneous functions of the scatter matrix are of interest. In line with this fact, scatter functionals often are only defined up to a positive scalar factor, yielding a family of scatter matrices rather than a uniquely defined one. In such families, it is natural to single out one representative by imposing a normalization constraint: this normalized scatter is called a shape matrix. In the particular case of elliptical families, this constraint in turn induces a concept of scale; along with a location center and a standardized radial density, the shape and scale parameters entirely characterize an elliptical density. In this paper, we show that one and only one normalization has the additional properties that (i) the resulting Fisher information matrices for shape and scale, in locally asymptotically normal (LAN) elliptical families, are block-diagonal, and that (ii) the semiparametric elliptical families indexed by location, shape, and completely unspecified radial densities are adaptive. This particular normalization, which imposes the condition that the determinant of the shape matrix should be equal to one, therefore can be considered canonical.

Suggested Citation

  • Paindaveine, Davy, 2008. "A canonical definition of shape," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2240-2247, October.
  • Handle: RePEc:eee:stapro:v:78:y:2008:i:14:p:2240-2247
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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. Marc Hallin & Bas Werker, 2003. "Semiparametric efficiency, distribution-freeness, and invariance," ULB Institutional Repository 2013/2119, ULB -- Universite Libre de Bruxelles.
    4. 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.
    5. Salibian-Barrera, Matias & Van Aelst, Stefan & Willems, Gert, 2006. "Principal Components Analysis Based on Multivariate MM Estimators With Fast and Robust Bootstrap," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1198-1211, September.
    6. Taskinen, Sara & Croux, Christophe & Kankainen, Annaliisa & Ollila, Esa & Oja, Hannu, 2006. "Influence functions and efficiencies of the canonical correlation and vector estimates based on scatter and shape matrices," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 359-384, February.
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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. Christophe Ley & Davy Paindaveine, 2010. "On Fisher information matrices and profile log-likelihood functions in generalized skew-elliptical models," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 235-250.
    3. 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.
    4. Dürre, Alexander & Vogel, Daniel, 2016. "Asymptotics of the two-stage spatial sign correlation," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 54-67.
    5. 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.
    6. Hallin, Marc & Šiman, Miroslav, 2016. "Elliptical multiple-output quantile regression and convex optimization," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 232-237.
    7. repec:bla:istatr:v:85:y:2017:i:1:p:108-142 is not listed on IDEAS
    8. 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.
    9. Davy Paindaveine & Germain Van Bever, 2017. "Tyler Shape Depth," Working Papers ECARES ECARES 2017-29, ULB -- Universite Libre de Bruxelles.
    10. Dürre, Alexander & Tyler, David E. & Vogel, Daniel, 2016. "On the eigenvalues of the spatial sign covariance matrix in more than two dimensions," Statistics & Probability Letters, Elsevier, vol. 111(C), pages 80-85.
    11. Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2014. "Efficient R-Estimation of Principal and Common Principal Components," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1071-1083, September.
    12. Frahm, Gabriel, 2009. "Asymptotic distributions of robust shape matrices and scales," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1329-1337, August.
    13. Marc Hallin & Miroslav Šiman, 2016. "Multiple-Output Quantile Regression," Working Papers ECARES ECARES 2016-03, ULB -- Universite Libre de Bruxelles.
    14. 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.
    15. Dürre, Alexander & Vogel, Daniel & Fried, Roland, 2015. "Spatial sign correlation," Journal of Multivariate Analysis, Elsevier, vol. 135(C), pages 89-105.
    16. 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.
    17. Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2009. "Optimal rank-based testing for principal component," Working Papers ECARES 2009_013, ULB -- Universite Libre de Bruxelles.
    18. Croux, C. & Dehon, C. & Yadine, A., 2010. "The K-Step Spatial Sign Covariance Matrix," Discussion Paper 2010-41, Tilburg University, Center for Economic Research.
    19. Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2011. "Optimal Rank-Based Tests for Common Principal Components," Working Papers ECARES ECARES 2011-032, ULB -- Universite Libre de Bruxelles.
    20. Marc Hallin, 2008. "On the Non Gaussian Asymptotics of the Likelihood Ratio Test Statistic for Homogeneity of Covariance," Working Papers ECARES 2008_039, ULB -- Universite Libre de Bruxelles.

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