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

Z. Bao, L. Lin, G. Pan, W. Zhou, Annals of Statistics 43 (2015) 2588–2623.


Journal Article | Published | English
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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.
Publishing Year
Date Published
2015-12-01
Journal Title
Annals of Statistics
Volume
43
Issue
6
Page
2588 - 2623
IST-REx-ID

Cite this

Bao Z, Lin L, Pan G, Zhou W. Spectral statistics of large dimensional spearman s rank correlation matrix and its application. Annals of Statistics. 2015;43(6):2588-2623. doi:10.1214/15-AOS1353
Bao, Z., Lin, L., Pan, G., & Zhou, W. (2015). Spectral statistics of large dimensional spearman s rank correlation matrix and its application. Annals of Statistics, 43(6), 2588–2623. https://doi.org/10.1214/15-AOS1353
Bao, Zhigang, Liang Lin, Guangming Pan, and Wang Zhou. “Spectral Statistics of Large Dimensional Spearman s Rank Correlation Matrix and Its Application.” Annals of Statistics 43, no. 6 (2015): 2588–2623. https://doi.org/10.1214/15-AOS1353.
Z. Bao, L. Lin, G. Pan, and W. Zhou, “Spectral statistics of large dimensional spearman s rank correlation matrix and its application,” Annals of Statistics, vol. 43, no. 6, pp. 2588–2623, 2015.
Bao Z, Lin L, Pan G, Zhou W. 2015. Spectral statistics of large dimensional spearman s rank correlation matrix and its application. Annals of Statistics. 43(6), 2588–2623.
Bao, Zhigang, et al. “Spectral Statistics of Large Dimensional Spearman s Rank Correlation Matrix and Its Application.” Annals of Statistics, vol. 43, no. 6, Institute of Mathematical Statistics, 2015, pp. 2588–623, doi:10.1214/15-AOS1353.

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