---
_id: '1504'
abstract:
- lang: eng
text: 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.
author:
- first_name: Zhigang
full_name: Bao, Zhigang
id: 442E6A6C-F248-11E8-B48F-1D18A9856A87
last_name: Bao
orcid: 0000-0003-3036-1475
- first_name: Liang
full_name: Lin, Liang
last_name: Lin
- first_name: Guangming
full_name: Pan, Guangming
last_name: Pan
- first_name: Wang
full_name: Zhou, Wang
last_name: Zhou
citation:
ama: 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
apa: 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*. Institute of Mathematical Statistics. https://doi.org/10.1214/15-AOS1353
chicago: 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*. Institute of Mathematical Statistics, 2015. https://doi.org/10.1214/15-AOS1353.
ieee: 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.
ista: 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.
mla: 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.
short: Z. Bao, L. Lin, G. Pan, W. Zhou, Annals of Statistics 43 (2015) 2588–2623.
date_created: 2018-12-11T11:52:24Z
date_published: 2015-12-01T00:00:00Z
date_updated: 2021-01-12T06:51:14Z
day: '01'
doi: 10.1214/15-AOS1353
extern: '1'
intvolume: ' 43'
issue: '6'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1312.5119
month: '12'
oa: 1
oa_version: Published Version
page: 2588 - 2623
publication: Annals of Statistics
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '5674'
quality_controlled: '1'
status: public
title: Spectral statistics of large dimensional spearman s rank correlation matrix
and its application
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 43
year: '2015'
...