Open Access
June 2019 Determinant of sample correlation matrix with application
Tiefeng Jiang
Ann. Appl. Probab. 29(3): 1356-1397 (June 2019). DOI: 10.1214/17-AAP1362

Abstract

Let x1,,xn be independent random vectors of a common p-dimensional normal distribution with population correlation matrix Rn. The sample correlation matrix R^n=(r^ij)p×p is generated from x1,,xn such that r^ij is the Pearson correlation coefficient between the ith column and the jth column of the data matrix (x1,,xn). The matrix R^n is a popular object in multivariate analysis and it has many connections to other problems. We derive a central limit theorem (CLT) for the logarithm of the determinant of R^n for a big class of Rn. The expressions of mean and the variance in the CLT are not obvious, and they are not known before. In particular, the CLT holds if p/n has a nonzero limit and the smallest eigenvalue of Rn is larger than 1/2. Besides, a formula of the moments of |R^n| and a new method of showing weak convergence are introduced. We apply the CLT to a high-dimensional statistical test.

Citation

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Tiefeng Jiang. "Determinant of sample correlation matrix with application." Ann. Appl. Probab. 29 (3) 1356 - 1397, June 2019. https://doi.org/10.1214/17-AAP1362

Information

Received: 1 October 2016; Revised: 1 August 2017; Published: June 2019
First available in Project Euclid: 19 February 2019

zbMATH: 07057457
MathSciNet: MR3914547
Digital Object Identifier: 10.1214/17-AAP1362

Subjects:
Primary: 60B20 , 60F05

Keywords: central limit theorem , moment generating function , multivariate normal distribution , sample correlation matrix , smallest eigenvalue

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.29 • No. 3 • June 2019
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