Signal detection in high dimensionthe multispiked case

Resumo

This paper applies Le Cam's asymptotic theory of statistical experiments to the signal detection problem in high dimension. We consider the problem of testing the null hypothesis of sphericity of a high-dimensional covariance matrix against an alternative of (unspecified) multiple symmetry-breaking directions (multispiked alternatives). Simple analytical expressions for the Gaussian asymptotic power envelope and the asymptotic powers of previously proposed tests are derived. Those asymptotic powers remain valid for non-Gaussian data satisfying mild moment restrictions. They appear to lie very substantially below the Gaussian power envelope, at least for small values of the number of symmetry-breaking directions. In contrast, the asymptotic power of Gaussian likelihood ratio tests based on the eigenvalues of the sample covariance matrix are shown to be very close to the envelope. Although based on Gaussian likelihoods, those tests remain valid under non-Gaussian densities satisfying mild moment conditions. The results of this paper extend to the case of multispiked alternatives and possibly non-Gaussian densities, the findings of an earlier study [Ann. Statist. 41 (2013) 1204-1231] of the single-spiked case. The methods we are using here, however, are entirely new, as the Laplace approximation methods considered in the single-spiked context do not extend to the multispiked case.

Ficha do documento

Tipo

Artigo científico

Ano

2014

Instituição

Inst Mathematical Statistics

Fonte

Repositório da FGV

Idioma

Inglês

Acesso

Acesso aberto

DOI

10.1214/13-AOS1181

Identificador

oai:repositorio.fgv.br:10438/23383

Palavras-chave

Sphericity testsLarge dimensionalityAsymptotic powerSpiked covarianceContiguityPower envelopeSample covariance matricesLargest eigenvalueWishartMatemáticaTeoria assintótica - Teoria da estimativaAnálise de variância