Spectral statistics of large dimensional Spearman’s rank correlation matrix and its application
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,...
| Main Authors: | , , , |
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| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
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2015
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| Online Access: | https://hdl.handle.net/10356/81031 http://hdl.handle.net/10220/39014 |
| _version_ | 1811681222825345024 |
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| author | Bao, Zhigang Lin, Liang-Ching Pan, Guangming Zhou, Wang |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Bao, Zhigang Lin, Liang-Ching Pan, Guangming Zhou, Wang |
| author_sort | Bao, Zhigang |
| collection | NTU |
| description | 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/√pZ 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. |
| first_indexed | 2024-10-01T03:37:31Z |
| format | Journal Article |
| id | ntu-10356/81031 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T03:37:31Z |
| publishDate | 2015 |
| record_format | dspace |
| spelling | ntu-10356/810312023-02-28T19:31:13Z Spectral statistics of large dimensional Spearman’s rank correlation matrix and its application Bao, Zhigang Lin, Liang-Ching Pan, Guangming Zhou, Wang School of Physical and Mathematical Sciences 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/√pZ 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. Published version 2015-12-09T08:11:35Z 2019-12-06T14:19:56Z 2015-12-09T08:11:35Z 2019-12-06T14:19:56Z 2015 Journal Article Bao, Z., Lin, L.-C., Pan, G., & Zhou, W. (2015). Spectral statistics of large dimensional Spearman’s rank correlation matrix and its application. The Annals of Statistics, 43(6), 2588-2623. 0090-5364 https://hdl.handle.net/10356/81031 http://hdl.handle.net/10220/39014 10.1214/15-AOS1353 en The Annals of Statistics © 2015 Institute of Mathematical Statistics. This paper was published in The Annals of Statistics and is made available as an electronic reprint (preprint) with permission of Institute of Mathematical Statistics. The published version is available at: [http://dx.doi.org/10.1214/15-AOS1353]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law. 37 p. application/pdf |
| spellingShingle | Bao, Zhigang Lin, Liang-Ching Pan, Guangming Zhou, Wang Spectral statistics of large dimensional Spearman’s rank correlation matrix and its application |
| title | Spectral statistics of large dimensional Spearman’s rank correlation matrix and its application |
| title_full | Spectral statistics of large dimensional Spearman’s rank correlation matrix and its application |
| title_fullStr | Spectral statistics of large dimensional Spearman’s rank correlation matrix and its application |
| title_full_unstemmed | Spectral statistics of large dimensional Spearman’s rank correlation matrix and its application |
| title_short | Spectral statistics of large dimensional Spearman’s rank correlation matrix and its application |
| title_sort | spectral statistics of large dimensional spearman s rank correlation matrix and its application |
| url | https://hdl.handle.net/10356/81031 http://hdl.handle.net/10220/39014 |
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