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Johnstone, I. M. and Onatski, A.

Testing in High-Dimensional Spiked Models


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 {code} 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

JEL Codes: E20 H15

Author links: Alexey Onatskiy  


Open Access Link:

Published Version of Paper: Testing in High-dimensional Spiked Models, Johnstone, I. M. and Onatski, A., Annals of Statistics, forthcoming (2018)

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