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November 2013 Optimal hypothesis testing for high dimensional covariance matrices
T. Tony Cai, Zongming Ma
Bernoulli 19(5B): 2359-2388 (November 2013). DOI: 10.3150/12-BEJ455

Abstract

This paper considers testing a covariance matrix $\Sigma$ in the high dimensional setting where the dimension $p$ can be comparable or much larger than the sample size $n$. The problem of testing the hypothesis $H_{0}:\Sigma=\Sigma_{0}$ for a given covariance matrix $\Sigma_{0}$ is studied from a minimax point of view. We first characterize the boundary that separates the testable region from the non-testable region by the Frobenius norm when the ratio between the dimension $p$ over the sample size $n$ is bounded. A test based on a $U$-statistic is introduced and is shown to be rate optimal over this asymptotic regime. Furthermore, it is shown that the power of this test uniformly dominates that of the corrected likelihood ratio test (CLRT) over the entire asymptotic regime under which the CLRT is applicable. The power of the $U$-statistic based test is also analyzed when $p/n$ is unbounded.

Citation

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T. Tony Cai. Zongming Ma. "Optimal hypothesis testing for high dimensional covariance matrices." Bernoulli 19 (5B) 2359 - 2388, November 2013. https://doi.org/10.3150/12-BEJ455

Information

Published: November 2013
First available in Project Euclid: 3 December 2013

zbMATH: 1281.62140
MathSciNet: MR3160557
Digital Object Identifier: 10.3150/12-BEJ455

Keywords: Correlation matrix , Covariance matrix , High-dimensional data , likelihood ratio test , minimax hypothesis testing , power , testing covariance structure

Rights: Copyright © 2013 Bernoulli Society for Mathematical Statistics and Probability

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Vol.19 • No. 5B • November 2013
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