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Influence functions and efficiencies of the canonical correlation and vector estimates based on scatter and shape matrices

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  • Taskinen, Sara
  • Croux, Christophe
  • Kankainen, Annaliisa
  • Ollila, Esa
  • Oja, Hannu

Abstract

In this paper, the influence functions and limiting distributions of the canonical correlations and coefficients based on affine equivariant scatter matrices are developed for elliptically symmetric distributions. General formulas for limiting variances and covariances of the canonical correlations and canonical vectors based on scatter matrices are obtained. Also the use of the so-called shape matrices in canonical analysis is investigated. The scatter and shape matrices based on the affine equivariant Sign Covariance Matrix as well as the Tyler's shape matrix serve as examples. Their finite sample and limiting efficiencies are compared to those of the Minimum Covariance Determinant estimators and S-estimator through theoretical and simulation studies. The theory is illustrated by an example.

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  • 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.
    • Handle: RePEc:eee:jmvana:v:97:y:2006:i:2:p:359-384
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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. Eaton, M. L. & Tyler, D., 1994. "The Asymptotic Distribution of Singular-Values with Applications to Canonical Correlations and Correspondence Analysis," Journal of Multivariate Analysis, Elsevier, vol. 50(2), pages 238-264, August.
    3. Croux, Christophe & Haesbroeck, Gentiane, 1999. "Influence Function and Efficiency of the Minimum Covariance Determinant Scatter Matrix Estimator," Journal of Multivariate Analysis, Elsevier, vol. 71(2), pages 161-190, November.
    4. Anderson, T. W., 1999. "Asymptotic Theory for Canonical Correlation Analysis," Journal of Multivariate Analysis, Elsevier, vol. 70(1), pages 1-29, July.
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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. Alvarez, Agustín & Boente, Graciela & Kudraszow, Nadia, 2019. "Robust sieve estimators for functional canonical correlation analysis," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 46-62.
    3. 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.
    4. 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.
    5. Jaakko Nevalainen & Denis Larocque & Hannu Oja, 2007. "A weighted spatial median for clustered data," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 15(3), pages 355-379, February.
    6. Langworthy, Benjamin W. & Stephens, Rebecca L. & Gilmore, John H. & Fine, Jason P., 2021. "Canonical correlation analysis for elliptical copulas," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    7. 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.
    8. Jorge G. Adrover & Stella M. Donato, 2023. "Aspects of robust canonical correlation analysis, principal components and association," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(2), pages 623-650, June.
    9. Jaakko Nevalainen & Denis Larocque & Hannu Oja, 2007. "A weighted spatial median for clustered data," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 15(3), pages 355-379, February.
    10. Ella Roelant & Stefan Aelst & Gert Willems, 2009. "The minimum weighted covariance determinant estimator," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 70(2), pages 177-204, September.
    11. Frahm, Gabriel, 2009. "Asymptotic distributions of robust shape matrices and scales," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1329-1337, August.
    12. Zhou, Jianhui, 2009. "Robust dimension reduction based on canonical correlation," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 195-209, January.
    13. Yamada, Tomoya, 2013. "Asymptotic properties of canonical correlation analysis for one group with additional observations," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 389-401.
    14. 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.
    15. Dürre, Alexander & Vogel, Daniel & Fried, Roland, 2015. "Spatial sign correlation," Journal of Multivariate Analysis, Elsevier, vol. 135(C), pages 89-105.
    16. Adrover, Jorge G. & Donato, Stella M., 2015. "A robust predictive approach for canonical correlation analysis," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 356-376.
    17. 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.
    18. Cator, Eric A. & Lopuhaä, Hendrik P., 2010. "Asymptotic expansion of the minimum covariance determinant estimators," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2372-2388, November.
    19. Bernard, Gaspard & Verdebout, Thomas, 2024. "On testing the equality of latent roots of scatter matrices under ellipticity," Journal of Multivariate Analysis, Elsevier, vol. 199(C).
    20. Paindaveine, Davy, 2008. "A canonical definition of shape," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2240-2247, October.

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