Signal Detection in High Dimension: The Multispiked Case

TL;DR

This paper analyzes the limits of detecting multiple signals in high-dimensional covariance matrices, showing that traditional tests underperform compared to the likelihood ratio test, which approaches optimal power.

Contribution

It extends the asymptotic analysis of signal detection from single-spiked to multispiked cases using novel Laplace approximation methods.

Findings

Likelihood ratio test approaches the power envelope.

Existing tests have substantially lower power.

Results extend previous single-spike findings to multiple spikes.

Abstract

This paper deals with the local asymptotic structure, in the sense of Le Cam's asymptotic theory of statistical experiments, of the signal detection problem in high dimension. More precisely, 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 (\textit{multispiked} alternatives). Simple analytical expressions for the asymptotic power envelope and the asymptotic powers of previously proposed tests are derived. These asymptotic powers are shown to lie very substantially below the envelope, at least for relatively small values of the number of symmetry-breaking directions under the alternative. In contrast, the asymptotic power of the likelihood ratio test based on the eigenvalues of the sample covariance matrix is shown to be close to that envelope. These results extend to the case of multispiked alternatives the findings of an earlier study (Onatski, Moreira and Hallin, 2011) of the single-spiked case. The methods we are using here, however, are entirely new, as the Laplace approximations considered in the single-spiked context do not extend to the multispiked case.