The Annals of Applied Probability

Determinant of sample correlation matrix with application

Tiefeng Jiang

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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.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 3 (2019), 1356-1397.

Dates
Received: October 2016
Revised: August 2017
First available in Project Euclid: 19 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1550566833

Digital Object Identifier
doi:10.1214/17-AAP1362

Mathematical Reviews number (MathSciNet)
MR3914547

Zentralblatt MATH identifier
07057457

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60F05: Central limit and other weak theorems

Keywords
Central limit theorem sample correlation matrix smallest eigenvalue multivariate normal distribution moment generating function

Citation

Jiang, Tiefeng. Determinant of sample correlation matrix with application. Ann. Appl. Probab. 29 (2019), no. 3, 1356--1397. doi:10.1214/17-AAP1362. https://projecteuclid.org/euclid.aoap/1550566833


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