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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.

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  • 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. 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 327-350, December.
    2. 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.
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    4. 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.
    5. Thomas P. Hettmansperger, 2002. "A practical affine equivariant multivariate median," Biometrika, Biometrika Trust, vol. 89(4), pages 851-860, December.
    6. Marc Hallin & Bas Werker, 2003. "Semiparametric efficiency, distribution-freeness, and invariance," ULB Institutional Repository 2013/2119, ULB -- Universite Libre de Bruxelles.
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