Signal Detection in High Dmension: The Multispiked Case

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.