--- res: bibo_abstract: - Let Q = (Q1, . . . , 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 Zj is the mean zero variance one random variable obtained by centralizing and normalizing Qj , j = 1, . . . , n. Assume that Xi , i = 1, . . . ,p are i.i.d. copies of 1/√ p Z and X = Xp,n is the p × n random matrix with Xi as its ith row. Then Sn = XX is called the p × 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 ∈ (0,∞) as n→∞.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.@eng bibo_authorlist: - foaf_Person: foaf_givenName: Zhigang foaf_name: Bao, Zhigang foaf_surname: Bao foaf_workInfoHomepage: http://www.librecat.org/personId=442E6A6C-F248-11E8-B48F-1D18A9856A87 orcid: 0000-0003-3036-1475 - foaf_Person: foaf_givenName: Liang foaf_name: Lin, Liang foaf_surname: Lin - foaf_Person: foaf_givenName: Guangming foaf_name: Pan, Guangming foaf_surname: Pan - foaf_Person: foaf_givenName: Wang foaf_name: Zhou, Wang foaf_surname: Zhou bibo_doi: 10.1214/15-AOS1353 bibo_issue: '6' bibo_volume: 43 dct_date: 2015^xs_gYear dct_language: eng dct_publisher: Institute of Mathematical Statistics@ dct_title: Spectral statistics of large dimensional spearman s rank correlation matrix and its application@ ...