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Asymptotic Power of Sphericity Tests for High-Dimensional Data

Author

Listed:
  • Alexei Onatski
  • Marcelo Moreira J.
  • Marc Hallin

Abstract

This paper studies the asymptotic power of tests of sphericity against perturbations in a single unknown direction as both the dimensionality of the data and the number of observations go to infinity. We establish the convergence, under the null hypothesis and the alternative, of the log ratio of the joint densities of the sample covariance eigenvalues to a Gaussian process indexed by the norm of the perturbation. When the perturbation norm is larger than the phase transition threshold studied in Baik et al. (2005), the limiting process is degenerate and discrimination between the null and the alternative is asymptotically certain. When the norm is below the threshold, the process is non-degenerate, so that the joint eigenvalue densities under the null and alternative hypotheses are mutually contiguous. Using the asymptotic theory of statistical experiments, we obtain asymptotic power envelopes and derive the asymptotic power for various sphericity tests in the contiguity region. In particular, we show that the asymptotic power of the Tracy-Widom-type tests is trivial, whereas that of the eigenvalue-based likelihood ratio test is strictly larger than the size, and close to the power envelope.

Suggested Citation

  • Alexei Onatski & Marcelo Moreira J. & Marc Hallin, 2011. "Asymptotic Power of Sphericity Tests for High-Dimensional Data," Working Papers ECARES ECARES 2011-018, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2013/94952
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    Citations

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    Cited by:

    1. Onatski, Alexei, 2012. "Asymptotics of the principal components estimator of large factor models with weakly influential factors," Journal of Econometrics, Elsevier, vol. 168(2), pages 244-258.
    2. Laurent Gobillon & Thierry Magnac, 2016. "Regional Policy Evaluation: Interactive Fixed Effects and Synthetic Controls," The Review of Economics and Statistics, MIT Press, vol. 98(3), pages 535-551, July.
    3. Laurent Gobillon & Thierry Magnac, 2016. "Regional Policy Evaluation: Interactive Fixed Effects and Synthetic Controls," The Review of Economics and Statistics, MIT Press, vol. 98(3), pages 535-551, July.
    4. Muni S. Srivastava & Hirokazu Yanagihara & Tatsuya Kubokawa, 2014. "Tests for Covariance Matrices in High Dimension with Less Sample Size," CIRJE F-Series CIRJE-F-933, CIRJE, Faculty of Economics, University of Tokyo.
    5. Marc Hallin & Marcelo Moreira J. & Alexei Onatski, 2013. "Group Invariance, Likelihood Ratio Tests, and the Incidental Parameter Problem in a High-Dimensional Linear Model," Working Papers ECARES ECARES 2013-04, ULB -- Universite Libre de Bruxelles.
    6. Moreira, Humberto Ataíde & Moreira, Marcelo J., 2015. "Optimal two-sided tests for instrumental variables regression with heteroskedastic and autocorrelated errors," FGV/EPGE Economics Working Papers (Ensaios Economicos da EPGE) 764, FGV/EPGE - Escola Brasileira de Economia e Finanças, Getulio Vargas Foundation (Brazil).
    7. Laurent Gobillon & François-Charles Wolff, 2015. "Évaluer l’effet des politiques publiques locales avec les contrôles synthétiques et les modèles à facteurs : Une application au marché du poisson français," PSE Working Papers halshs-01183455, HAL.
    8. Alexei Onatski & Marcelo Moreira J. & Marc Hallin, 2012. "Signal Detection in High Dmension: The Multispiked Case," Working Papers ECARES ECARES 2012-036, ULB -- Universite Libre de Bruxelles.
    9. Davy Paindaveine & Thomas Verdebout, 2013. "Universal Asymptotics for High-Dimensional Sign Tests," Working Papers ECARES ECARES 2013-40, ULB -- Universite Libre de Bruxelles.

    More about this item

    Keywords

    sphericity tests; large dimentionality; asymptotic power; spiker covariance; contiguity; power enveloppe; steepest descent; contour intgral representation;

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