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Efficient R-Estimation of Principal and Common Principal Components


  • Marc Hallin
  • Davy Paindaveine
  • Thomas Verdebout


We propose rank-based estimators of principal components, both in the one-sample and, under the assumption of common principal components , in the m -sample cases. Those estimators are obtained via a rank-based version of Le Cam's one-step method, combined with an estimation of cross-information quantities . Under arbitrary elliptical distributions with, in the m -sample case, possibly heterogeneous radial densities, those R-estimators remain root- n consistent and asymptotically normal, while achieving asymptotic efficiency under correctly specified radial densities. Contrary to their traditional counterparts computed from empirical covariances, they do not require any moment conditions. When based on Gaussian score functions, in the one-sample case, they uniformly dominate their classical competitors in the Pitman sense. Their AREs with respect to other robust procedures are quite high-up to 30, in the Gaussian case, with respect to minimum covariance determinant estimators. Their finite-sample performances are investigated via a Monte Carlo study.
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  • Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2013. "Efficient R-Estimation of Principal and Common Principal Components," Working Papers ECARES ECARES 2013-18, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2013/142830

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    References listed on IDEAS

    1. Boente, Graciela & Pires, Ana M. & Rodrigues, Isabel M., 2006. "General projection-pursuit estimators for the common principal components model: influence functions and Monte Carlo study," Journal of Multivariate Analysis, Elsevier, vol. 97(1), pages 124-147, January.
    2. Thomas P. Hettmansperger, 2002. "A practical affine equivariant multivariate median," Biometrika, Biometrika Trust, vol. 89(4), pages 851-860, December.
    3. Paindaveine, Davy, 2006. "A Chernoff-Savage result for shape:On the non-admissibility of pseudo-Gaussian methods," Journal of Multivariate Analysis, Elsevier, vol. 97(10), pages 2206-2220, November.
    4. Paindaveine, Davy, 2008. "A canonical definition of shape," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2240-2247, October.
    5. Boente, Graciela & Rodriguez, Daniela & Sued, Mariela, 2010. "Inference under functional proportional and common principal component models," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 464-475, February.
    6. 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.
    7. Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2009. "Optimal rank-based testing for principal component," Working Papers ECARES 2009_013, ULB -- Universite Libre de Bruxelles.
    8. 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.
    9. Croux, Christophe & Ruiz-Gazen, Anne, 2005. "High breakdown estimators for principal components: the projection-pursuit approach revisited," Journal of Multivariate Analysis, Elsevier, vol. 95(1), pages 206-226, July.
    10. Graciela Boente, 2002. "Influence functions and outlier detection under the common principal components model: A robust approach," Biometrika, Biometrika Trust, vol. 89(4), pages 861-875, December.
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    Cited by:

    1. Hallin, Marc & van den Akker, Ramon & Werker, Bas J.M., 2016. "Semiparametric error-correction models for cointegration with trends: Pseudo-Gaussian and optimal rank-based tests of the cointegration rank," Journal of Econometrics, Elsevier, vol. 190(1), pages 46-61.
    2. Christophe Ley & Yvik Swan & Thomas Verdebout, 2017. "Efficient ANOVA for directional data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(1), pages 39-62, February.
    3. Davy Paindaveine & Julien Remy & Thomas Verdebout, 2017. "Testing for Principal Component Directions under Weak Identifiability," Working Papers ECARES ECARES 2017-37, ULB -- Universite Libre de Bruxelles.
    4. Paindaveine, Davy & Rasoafaraniaina, Rondrotiana Joséa & Verdebout, Thomas, 2017. "Preliminary test estimation for multi-sample principal components," Econometrics and Statistics, Elsevier, vol. 2(C), pages 106-116.

    More about this item


    common principal components; elliptical densities; uniform local Asymptotic normality; principal components; ranks; R-estimation; robustness;

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