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

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/√pZ 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.Published versio