SPECTRAL STATISTICS OF LARGE DIMENSIONAL SPEARMAN'S RANK CORRELATION MATRIX AND ITS APPLICATION

• 成果类型: Article
• 署名作者: Bao, Zhigang; Lin, Liang-Ching; Pan, Guangming; Zhou, Wang
• 署名单位: Nanyang Technological University; National Cheng Kung University; National University of Singapore
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/15-AOS1353
• 发表日期: 2015
• 页码: 2588-2623
• 关键词: limit-theorems; sample; INDEPENDENCE; tests; ratio

摘要

Let Q = (Qi,...,Qn) be a random vector drawn from the uniform distribution on the set of all n! permutations of {1,2,...,n}. Let Z = (Z1,...,Zn), where Z(j) is the mean zero variance one random variable obtained by centralizing and normalizing Q(j), j = 1,...,n. Assume that X-i, i = 1,...,p are i.i.d. copies of 1 root p Z and X = Xp,n is the p x n random matrix with X-i as its ith row. Then S-n = XX* is called the p x 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 -> c is an element of (0, infinity) as n -> infinity. 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.