Spectral statistics of large dimensional spearman s rank correlation matrix and its application Journal Article

Author(s): Bao, Zhigang; Lin, Liang-Ching; Pan, Guangming; Zhou, Wang
Article Title: Spectral statistics of large dimensional spearman s rank correlation matrix and its application
Affiliation IST Austria
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.
Keywords: Central limit theorem; Independence test; Linear spectral statistics; Nonparametric method; Spearman's rank correlation matrix
Journal Title: Annals of Statistics
Volume: 43
Issue 6
ISSN: 0090-5364
Publisher: Institute of Mathematical Statistics  
Date Published: 2015-12-01
Start Page: 2588
End Page: 2623
DOI: 10.1214/15-AOS1353
Notes: B.Z. was supported in part by the Ministry of Education, Singapore, under Grant # ARC 14/11, and NSF of China, Grant No. 11371317. L.L.was supported in part by a MOE Tier 2 Grant 2014-T2-2-060 and by a MOE Tier 1 Grant RG25/14 at the Nanyang Technological University, Singapore. G.P. was supported in part by Grant R-155-000-151-112 at the National University of Singapore.
Open access: yes (repository)