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

Abstract

Let $\mathbf{Q}=(Q_1,\dots ,Q_n)$ be a random vector drawn from the uniform distribution on the set of all $n!$ permutations of $\{1,2,\dots ,n\}$. Let $\mathbf{Z}=(Z_1,\dots ,Z_n)$, where $Z_j$ is the mean zero variance one random variable obtained by centralizing and normalizing $Q_j$, $j=1,\dots ,n$. Assume that ${\mathbf{X}}_i,i=1,\dots ,p$ 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 $S_n=X{X}^{\ast }$ 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.

Published at http://dx.doi.org/10.1214/15-AOS1353 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Central limit theorem for signal-to-interference ratio of reduced rank linear receiver
• A CLT for regularized sample covariance matrices

Cited by in corpus (6)

• Limiting spectral distribution of large dimensional Spearman's rank correlation matrices
• Joint limiting laws for high-dimensional independence tests
• An Extreme-Value Approach for Testing the Equality of Large U-Statistic Based Correlation Matrices
• Linear Eigenvalue Statistics of $X{X}^{\prime }$ matrices
• Linear spectral statistics of sequential sample covariance matrices
• Tracy-Widom limit for Kendall's tau