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Testing in high-dimensional spiked models

Accepted version
Peer-reviewed

Type

Article

Change log

Authors

Johnstone, IM 
Onatski, A 

Abstract

We consider the five classes of multivariate statistical problems identified by James (1964), which together cover much of classical multivariate analysis, plus a simpler limiting case, symmetric matrix denoising. Each of James' problems involves the eigenvalues of E−1H where H and E are proportional to high dimensional Wishart matrices. Under the null hypothesis, both Wisharts are central with identity covariance. Under the alternative, the non-centrality or the covariance parameter of H has a single eigenvalue, a spike, that stands alone. When the spike is smaller than a case-specific phase transition threshold, none of the sample eigenvalues separate from the bulk, making the testing problem challenging. Using a unified strategy for the six cases, we show that the log likelihood ratio processes parameterized by the value of the sub-critical spike converge to Gaussian processes with logarithmic correlation. We then derive asymptotic power envelopes for tests for the presence of a spike.

Description

Keywords

math.ST, math.ST, stat.TH, 62H15, 62F05

Journal Title

Annals of Statistics

Conference Name

Journal ISSN

0090-5364
2168-8966

Volume Title

48

Publisher

Institute of Mathematical Statistics