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June 2013 Asymptotic power of sphericity tests for high-dimensional data
Alexei Onatski, Marcelo J. Moreira, Marc Hallin
Ann. Statist. 41(3): 1204-1231 (June 2013). DOI: 10.1214/13-AOS1100

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 contiguous alternatives, 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, Ben Arous and Péché [Ann. Probab. 33 (2005) 1643–1697] the limiting process is degenerate, and discrimination between the null and the alternative is asymptotically certain. When the norm is below the threshold, the limiting process is nondegenerate, and 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 (i.e., equals the asymptotic size), 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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Alexei Onatski. Marcelo J. Moreira. Marc Hallin. "Asymptotic power of sphericity tests for high-dimensional data." Ann. Statist. 41 (3) 1204 - 1231, June 2013. https://doi.org/10.1214/13-AOS1100

Information

Published: June 2013
First available in Project Euclid: 13 June 2013

zbMATH: 1293.62125
MathSciNet: MR3113808
Digital Object Identifier: 10.1214/13-AOS1100

Subjects:
Primary: 62B15 , 62H15
Secondary: 41A60

Keywords: asymptotic power , contiguity , contour integral representation , large dimensionality , power envelope , Sphericity tests , spiked covariance , steepest descent

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 3 • June 2013
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