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Asymptotic Power of Sphericity Tests for High-Dimensional Data

  • Alexei Onatski
  • Marcelo Moreira J.
  • Marc Hallin

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.

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File URL: https://dipot.ulb.ac.be/dspace/bitstream/2013/94952/1/2011-018-ONATSKI_MOREIRA_HALLIN-asymptotic.pdf
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Paper provided by ULB -- Universite Libre de Bruxelles in its series Working Papers ECARES with number ECARES 2011-018.

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Length: 61 p.
Date of creation: Aug 2011
Date of revision:
Publication status: Published by:
Handle: RePEc:eca:wpaper:2013/94952
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