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Signal Detection in High Dmension: The Multispiked Case

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  • Alexei Onatski
  • Marcelo Moreira J.
  • Marc Hallin

Abstract

This paper deals with the local asymptotic structure, in the sense ofLe Cam’s asymptotic theory of statistical experiments, of the signal detectionproblem in high dimension. More precisely, we consider the problemof testing the null hypothesis of sphericity of a high-dimensional covariancematrix against an alternative of (unspecified) multiple symmetry-breakingdirections (multispiked alternatives). Simple analytical expressions for theasymptotic power envelope and the asymptotic powers of previously proposedtests are derived. These asymptotic powers are shown to lie verysubstantially below the envelope, at least for relatively small values of thenumber of symmetry-breaking directions under the alternative. In contrast,the asymptotic power of the likelihood ratio test based on the eigenvalues ofthe sample covariance matrix is shown to be close to that envelope. Theseresults extend to the case of multispiked alternatives the findings of an earlierstudy (Onatski, Moreira and Hallin, 2011) of the single-spiked case. The methods we are using here, however, are entirely new, as the Laplace approximationsconsidered in the single-spiked context do not extend to themultispiked case.

Suggested Citation

  • 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.
  • Handle: RePEc:eca:wpaper:2013/130318
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    References listed on IDEAS

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    1. 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.
    2. Schott, James R., 2006. "A high-dimensional test for the equality of the smallest eigenvalues of a covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 97(4), pages 827-843, April.
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    Cited by:

    1. Matteo Barigozzi & Lorenzo Trapani, 2017. "Sequential testing for structural stability in approximate factor models," Papers 1708.02786, arXiv.org, revised Mar 2018.

    More about this item

    Keywords

    sphericity tests; large dimentionality; asymptotic power; spiked covariance; contiguity; power enveloppe;

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