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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 $\mathbf{x}_{1},\ldots ,\mathbf{x}_{n}$ be independent random vectors of a common $p$-dimensional normal distribution with population correlation matrix $\mathbf{R}_{n}$. The sample correlation matrix $\hat{\mathbf {R}}_{n}=(\hat{r}_{ij})_{p\times p}$ is generated from $\mathbf{x}_{1},\ldots ,\mathbf{x}_{n}$ such that $\hat{r}_{ij}$ is the Pearson correlation coefficient between the $i$th column and the $j$th column of the data matrix $(\mathbf{x}_{1},\ldots ,\mathbf{x}_{n})'$. The matrix $\hat{\mathbf {R}}_{n}$ 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 $\hat{\mathbf {R}}_{n}$ for a big class of $\mathbf{R}_{n}$. 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 $\mathbf{R}_{n}$ is larger than $1/2$. Besides, a formula of the moments of $\vert \hat{\mathbf {R}}_{n}\vert $ 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

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 3 • June 2019
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