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Optimal Rank-Based Tests for Common Principal Components


  • Marc Hallin
  • Davy Paindaveine
  • Thomas Verdebout


This paper provides optimal testing procedures for the m-sample null hypothesis of Common Principal Components (CPC) under possibly non Gaussian and heterogenous elliptical densities. We first establish, under very mild assumptions that do not require finite moments of order four, the local asymptotic normality (LAN) of the model. Based on that result, we show that the pseudo-Gaussian test proposed in Hallin et al. (2010a) is locally and asymptotically optimal under Gaussian densities. We also show how to compute its local powers and asymptotic relative efficiencies (AREs). A numerical evaluation of those AREs, however, reveals that, while remaining valid, this test is poorly efficient away from the Gaussian. Moreover, it still requires finite moments of order four. We therefore propose rank-based procedures that remain valid under any possibly heterogenous m-tuple of elliptical densities, irrespective of any moment assumptions—in elliptical families, indeed, principal components naturally can be based on the scatter matrices characterizing the density contours, hence do not require finite variances. Those rank-based tests are not only validity-robust in the sense that they survive arbitrary elliptical population densities: we show that they also are efficiency-robust, in the sense that their local powers do not deteriorate under non-Gaussian alternatives. In the homogeneous case, the normal-score version of our tests uniformly dominates, in the Pitman sense, the optimal pseudo-Gaussian test. Theoretical results are obtained via a nonstandard application of Le Cam’s methodology in the context of curved LAN experiments. The finite-sample properties of the proposed tests are investigated through simulations

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  • 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.
  • Handle: RePEc:eca:wpaper:2013/101786

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    References listed on IDEAS

    1. Marc Hallin & Bas Werker, 2003. "Semiparametric efficiency, distribution-freeness, and invariance," ULB Institutional Repository 2013/2119, ULB -- Universite Libre de Bruxelles.
    2. Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2009. "Optimal rank-based testing for principal component," Working Papers ECARES 2009_013, ULB -- Universite Libre de Bruxelles.
    3. Robert J. Boik, 2002. "Spectral models for covariance matrices," Biometrika, Biometrika Trust, vol. 89(1), pages 159-182, March.
    4. 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.
    5. Paindaveine, Davy, 2008. "A canonical definition of shape," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2240-2247, October.
    6. 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.
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    Cited by:

    1. Hallin, M. & van den Akker, R. & Werker, B.J.M., 2012. "Rank-based Tests of the Cointegrating Rank in Semiparametric Error Correction Models," Discussion Paper 2012-089, Tilburg University, Center for Economic Research.
    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. Hallin, M. & Werker, B.J.M. & van den Akker, R., 2015. "Optimal Pseudo-Gaussian and Rank-based Tests of the Cointegration Rank in Semiparametric Error-correction Models," Discussion Paper 2015-001, Tilburg University, Center for Economic Research.
    4. 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.

    More about this item


    Common Principal Components; Rank-Based Methods; Local Asymptotic Normality; Robustness;

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