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February 2018 Testing independence in high dimensions with sums of rank correlations
Dennis Leung, Mathias Drton
Ann. Statist. 46(1): 280-307 (February 2018). DOI: 10.1214/17-AOS1550

Abstract

We treat the problem of testing independence between $m$ continuous variables when $m$ can be larger than the available sample size $n$. We consider three types of test statistics that are constructed as sums or sums of squares of pairwise rank correlations. In the asymptotic regime where both $m$ and $n$ tend to infinity, a martingale central limit theorem is applied to show that the null distributions of these statistics converge to Gaussian limits, which are valid with no specific distributional or moment assumptions on the data. Using the framework of U-statistics, our result covers a variety of rank correlations including Kendall’s tau and a dominating term of Spearman’s rank correlation coefficient (rho), but also degenerate U-statistics such as Hoeffding’s $D$, or the $\tau^{*}$ of Bergsma and Dassios [Bernoulli 20 (2014) 1006–1028]. As in the classical theory for U-statistics, the test statistics need to be scaled differently when the rank correlations used to construct them are degenerate U-statistics. The power of the considered tests is explored in rate-optimality theory under a Gaussian equicorrelation alternative as well as in numerical experiments for specific cases of more general alternatives.

Citation

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Dennis Leung. Mathias Drton. "Testing independence in high dimensions with sums of rank correlations." Ann. Statist. 46 (1) 280 - 307, February 2018. https://doi.org/10.1214/17-AOS1550

Information

Received: 1 October 2015; Revised: 1 December 2016; Published: February 2018
First available in Project Euclid: 22 February 2018

zbMATH: 06865112
MathSciNet: MR3766953
Digital Object Identifier: 10.1214/17-AOS1550

Subjects:
Primary: 60K35

Keywords: High-dimensional statistics , independence , Minimax optimality , rank correlations , U-statistics

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 1 • February 2018
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