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Testing for Principal Component Directions under Weak Identifiability


  • Davy Paindaveine
  • Julien Remy
  • Thomas Verdebout


We consider the problem of testing, on the basis of a p-variate Gaussian random sample, the null hypothesis H_0: \theta_1= \theta_1^0 against the alternative H_1: \theta_1 \neq \thetab_1^0, where \thetab_1 is the "first" eigenvector of the underlying covariance matrix and \thetab_1^0 is a fixed unit p-vector. In the classical setup where eigenvalues \lambda_1>\lambda_2\geq .\geq \lambda_p are fixed, the Anderson (1963) likelihood ratio test (LRT) and the Hallin, Paindavine and Verdebout (2010) Le Cam optimal test for this problem are asymptotically equivalent under the null, hence also under sequences of contiguous alternatives. We show that this equivalence does not survive asymptotic scenarios where \lambda_{n1}-\lambda_{n2}=o(r_n) with r_n=O(1/\sqrt{n}). For such scenarios, the Le Cam optimal test still asymptotically meets the nominal level constraint, whereas the LRT becomes extremely liberal. Consequently, the former test should be favored over the latter one whenever the two largest sample eigenvalues are close to each other. By relying on the Le Cam theory of asymptotic experiments, we study in the aforementioned asymptotic scenarios the non-null and optimality properties of the Le Cam optimal test and show that the null robustness of this test is not obtained at the expense of efficiency. Our asymptotic investigation is extensive in the sense that it allows r_n to converge to zero at an arbitrary rate. To make our results as striking as possible, we not only restrict to the multinormal case but also to single-spiked spectra of the form \lambda_{n1}>\lambda_{n2}=.=\lambda_{np}.

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  • 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.
  • Handle: RePEc:eca:wpaper:2013/259598

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    References listed on IDEAS

    1. Forchini, Giovanni & Hillier, Grant, 2003. "Conditional Inference For Possibly Unidentified Structural Equations," Econometric Theory, Cambridge University Press, vol. 19(05), pages 707-743, October.
    2. Johnstone, Iain M. & Lu, Arthur Yu, 2009. "On Consistency and Sparsity for Principal Components Analysis in High Dimensions," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 682-693.
    3. Dufour, Jean-Marie, 2006. "Monte Carlo tests with nuisance parameters: A general approach to finite-sample inference and nonstandard asymptotics," Journal of Econometrics, Elsevier, vol. 133(2), pages 443-477, August.
    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.
    5. Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2009. "Optimal rank-based testing for principal component," Working Papers ECARES 2009_013, ULB -- Universite Libre de Bruxelles.
    6. Fang Han & Han Liu, 2014. "Scale-Invariant Sparse PCA on High-Dimensional Meta-Elliptical Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(505), pages 275-287, March.
    7. Jean-Marie Dufour, 1997. "Some Impossibility Theorems in Econometrics with Applications to Structural and Dynamic Models," Econometrica, Econometric Society, vol. 65(6), pages 1365-1388, November.
    8. Boente, Graciela & Fraiman, Ricardo, 2000. "Kernel-based functional principal components," Statistics & Probability Letters, Elsevier, vol. 48(4), pages 335-345, July.
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