The Annals of Statistics

Testing independence with high-dimensional correlated samples

Xi Chen and Weidong Liu

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Testing independence among a number of (ultra) high-dimensional random samples is a fundamental and challenging problem. By arranging $n$ identically distributed $p$-dimensional random vectors into a $p\times n$ data matrix, we investigate the problem of testing independence among columns under the matrix-variate normal modeling of data. We propose a computationally simple and tuning-free test statistic, characterize its limiting null distribution, analyze the statistical power and prove its minimax optimality. As an important by-product of the test statistic, a ratio-consistent estimator for the quadratic functional of a covariance matrix from correlated samples is developed. We further study the effect of correlation among samples to an important high-dimensional inference problem—large-scale multiple testing of Pearson’s correlation coefficients. Indeed, blindly using classical inference results based on the assumed independence of samples will lead to many false discoveries, which suggests the need for conducting independence testing before applying existing methods. To address the challenge arising from correlation among samples, we propose a “sandwich estimator” of Pearson’s correlation coefficient by de-correlating the samples. Based on this approach, the resulting multiple testing procedure asymptotically controls the overall false discovery rate at the nominal level while maintaining good statistical power. Both simulated and real data experiments are carried out to demonstrate the advantages of the proposed methods.

Article information

Ann. Statist., Volume 46, Number 2 (2018), 866-894.

Received: November 2015
Revised: March 2017
First available in Project Euclid: 3 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F05: Asymptotic properties of tests
Secondary: 62H10: Distribution of statistics

Independence test multiple testing of correlations false discovery rate matrix-variate normal quadratic functional estimation high-dimensional sample correlation matrix


Chen, Xi; Liu, Weidong. Testing independence with high-dimensional correlated samples. Ann. Statist. 46 (2018), no. 2, 866--894. doi:10.1214/17-AOS1571.

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Supplemental materials

  • Supplement to “Testing independence with high-dimensional correlated samples”. We provide the proofs of all the theorectial results as well as additional simulated and real experimental results.