## The Annals of Statistics

### Spectral statistics of large dimensional Spearman’s rank correlation matrix and its application

#### 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.

#### Article information

Source
Ann. Statist., Volume 43, Number 6 (2015), 2588-2623.

Dates
Revised: June 2015
First available in Project Euclid: 7 October 2015

https://projecteuclid.org/euclid.aos/1444222086

Digital Object Identifier
doi:10.1214/15-AOS1353

Mathematical Reviews number (MathSciNet)
MR3405605

Zentralblatt MATH identifier
1328.15046

Subjects
Primary: 15B52: Random matrices
Secondary: 62H10: Distribution of statistics

#### Citation

Bao, Zhigang; Lin, Liang-Ching; Pan, Guangming; Zhou, Wang. Spectral statistics of large dimensional Spearman’s rank correlation matrix and its application. Ann. Statist. 43 (2015), no. 6, 2588--2623. doi:10.1214/15-AOS1353. https://projecteuclid.org/euclid.aos/1444222086

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#### Supplemental materials

• Supplement to “Spectral statistics of large dimensional Spearman’s rank correlation matrix and its application”. This supplemental article [5] contains the proofs of Lemmas 3.2, 4.1 4.5, 4.6, 4.11, Propositions 4.2, 4.15, Lemmas 5.1, 6.1, 6.2.