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June 2020 Testing in high-dimensional spiked models
Iain M. Johnstone, Alexei Onatski
Ann. Statist. 48(3): 1231-1254 (June 2020). DOI: 10.1214/18-AOS1697


We consider the five classes of multivariate statistical problems identified by James (Ann. Math. Stat. 35 (1964) 475–501), which together cover much of classical multivariate analysis, plus a simpler limiting case, symmetric matrix denoising. Each of James’ problems involves the eigenvalues of $E^{-1}H$ 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 noncentrality 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 subcritical spike converge to Gaussian processes with logarithmic correlation. We then derive asymptotic power envelopes for tests for the presence of a spike.


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Iain M. Johnstone. Alexei Onatski. "Testing in high-dimensional spiked models." Ann. Statist. 48 (3) 1231 - 1254, June 2020.


Received: 1 February 2017; Revised: 1 January 2018; Published: June 2020
First available in Project Euclid: 17 July 2020

zbMATH: 07241589
MathSciNet: MR4124321
Digital Object Identifier: 10.1214/18-AOS1697

Primary: 62E20
Secondary: 62H15

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.48 • No. 3 • June 2020
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