Zhigang Bao et al., SPECTRAL STATISTICS OF LARGE DIMENSIONAL SPEARMAN'S RANK CORRELATION MATRIX AND ITS APPLICATION, Annals of statistics , 43(6), 2015, pp. 2588-2623
Let Q = (Q. ,..., Qn) be a random vector drawn from the uniform distribution on the set of all n! permutations of {1, 2, ..., n]. Let Z = (Z. ,..., 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 $\frac{1}{{\sqrt p }}Z$ 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.