Asymptotic power of sphericity tests for high-dimensional data

Resumo

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 Peche [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.

Ficha do documento

Tipo

Artigo científico

Ano

2013

Instituição

Inst Mathematical Statistics

Fonte

Repositório da FGV

Idioma

Inglês

Acesso

Acesso aberto

DOI

10.1214/13-AOS1100

Identificador

oai:repositorio.fgv.br:10438/23332

Palavras-chave

Sphericity testsLarge dimensionalityAsymptotic powerSpiked covarianceContiguityPower envelopeSteepest descentContour integral representationSample covariance matricesMultiple hypergeometric-functionsEigenvalue based detectionEmpirical distributionDistributionsNumberSignalsModelsComponentsWishartMatemáticaTeoria assintótica - Teoria da estimativaDistribuição assintótica (Teoria da probabilidade)