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

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.

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Author

Bao, Zhigang

^{ISTA}^{}; Lin, Liang; Pan, Guangming; Zhou, WangAbstract

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-AOS1353Bao, Z., Lin, L., Pan, G., & Zhou, W. (2015). Spectral statistics of large dimensional spearman s rank correlation matrix and its application.

*Annals of Statistics*. Institute of Mathematical Statistics. https://doi.org/10.1214/15-AOS1353Bao, Zhigang, Liang Lin, Guangming Pan, and Wang Zhou. “Spectral Statistics of Large Dimensional Spearman s Rank Correlation Matrix and Its Application.”

*Annals of Statistics*. Institute of Mathematical Statistics, 2015. 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. Institute of Mathematical Statistics, pp. 2588–2623, 2015.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.**All files available under the following license(s):**

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