## Bernoulli

- Bernoulli
- Volume 23, Number 2 (2017), 1130-1178.

### CLT for eigenvalue statistics of large-dimensional general Fisher matrices with applications

Shurong Zheng, Zhidong Bai, and Jianfeng Yao

#### Abstract

Random Fisher matrices arise naturally in multivariate statistical analysis and understanding the properties of its eigenvalues is of primary importance for many hypothesis testing problems like testing the equality between two covariance matrices, or testing the independence between sub-groups of a multivariate random vector. Most of the existing work on random Fisher matrices deals with a particular situation where the population covariance matrices are equal. In this paper, we consider general Fisher matrices with arbitrary population covariance matrices and develop their spectral properties when the dimensions are proportionally large compared to the sample size. The paper has two main contributions: first the limiting distribution of the eigenvalues of a general Fisher matrix is found and second, a central limit theorem is established for a wide class of functionals of these eigenvalues. Applications of the main results are also developed for testing hypotheses on high-dimensional covariance matrices.

#### Article information

**Source**

Bernoulli, Volume 23, Number 2 (2017), 1130-1178.

**Dates**

Received: September 2014

Revised: April 2015

First available in Project Euclid: 4 February 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1486177395

**Digital Object Identifier**

doi:10.3150/15-BEJ772

**Mathematical Reviews number (MathSciNet)**

MR3606762

**Zentralblatt MATH identifier**

1375.60066

**Keywords**

central limit theorem equality of covariance matrices large-dimensional covariance matrices large-dimensional Fisher matrix linear spectral statistics

#### Citation

Zheng, Shurong; Bai, Zhidong; Yao, Jianfeng. CLT for eigenvalue statistics of large-dimensional general Fisher matrices with applications. Bernoulli 23 (2017), no. 2, 1130--1178. doi:10.3150/15-BEJ772. https://projecteuclid.org/euclid.bj/1486177395