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December 2015 Spectral statistics of large dimensional Spearman’s rank correlation matrix and its application
Zhigang Bao, Liang-Ching Lin, Guangming Pan, Wang Zhou
Ann. Statist. 43(6): 2588-2623 (December 2015). DOI: 10.1214/15-AOS1353

Abstract

Let $\mathbf{Q}=(Q_{1},\ldots,Q_{n})$ be a random vector drawn from the uniform distribution on the set of all $n!$ permutations of $\{1,2,\ldots,n\}$. Let $\mathbf{Z}=(Z_{1},\ldots,Z_{n})$, where $Z_{j}$ is the mean zero variance one random variable obtained by centralizing and normalizing $Q_{j}$, $j=1,\ldots,n$. Assume that $\mathbf{X}_{i},i=1,\ldots,p$ are i.i.d. copies of $\frac{1}{\sqrt{p}}\mathbf{Z}$ and $X=X_{p,n}$ is the $p\times n$ random matrix with $\mathbf{X}_{i}$ as its $i$th row. Then $S_{n}=XX^{*}$ is called the $p\times n$ Spearman’s rank correlation matrix which can be regarded as a high dimensional extension of the classical nonparametric statistic Spearman’s rank correlation coefficient between two independent random variables. In this paper, we establish a CLT for the linear spectral statistics of this nonparametric random matrix model in the scenario of high dimension, namely, $p=p(n)$ and $p/n\to c\in(0,\infty)$ as $n\to\infty$. We propose a novel evaluation scheme to estimate the core quantity in Anderson and Zeitouni’s cumulant method in [Ann. Statist. 36 (2008) 2553–2576] to bypass the so-called joint cumulant summability. In addition, we raise a two-step comparison approach to obtain the explicit formulae for the mean and covariance functions in the CLT. Relying on this CLT, we then construct a distribution-free statistic to test complete independence for components of random vectors. Owing to the nonparametric property, we can use this test on generally distributed random variables including the heavy-tailed ones.

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Zhigang Bao. Liang-Ching Lin. Guangming Pan. Wang Zhou. "Spectral statistics of large dimensional Spearman’s rank correlation matrix and its application." Ann. Statist. 43 (6) 2588 - 2623, December 2015. https://doi.org/10.1214/15-AOS1353

Information

Received: 1 October 2014; Revised: 1 June 2015; Published: December 2015
First available in Project Euclid: 7 October 2015

zbMATH: 1328.15046
MathSciNet: MR3405605
Digital Object Identifier: 10.1214/15-AOS1353

Subjects:
Primary: 15B52
Secondary: 62H10

Keywords: central limit theorem , Independence test , Linear spectral statistics , nonparametric method , Spearman’s rank correlation matrix

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 6 • December 2015
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