Johnstone, I. M. and Onatski, A.

Testing in High-dimensional Spiked Models

Annals of Statistics

Vol. 48 no. 3 pp. 1231-1254 (2018)

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 H -1 E 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.

Keywords: likelihood ratio test, hypergeometric function, principal components analysis, canonical correlations, matrix denoising, multiple response regression

Author links: Alexey Onatskiy

Publisher's Link: https://doi.org/10.1214/18-AOS1697