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