## The Annals of Statistics

- Ann. Statist.
- Volume 37, Number 6B (2009), 3822-3840.

### Corrections to LRT on large-dimensional covariance matrix by RMT

Zhidong Bai, Dandan Jiang, Jian-Feng Yao, and Shurong Zheng

#### Abstract

In this paper, we give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from Gaussian populations when the dimension *p* is large compared to the sample size *n*. Next, using recent central limit theorems for linear spectral statistics of sample covariance matrices and of random *F*-matrices, we propose necessary corrections for these LR tests to cope with high-dimensional effects. The asymptotic distributions of these corrected tests under the null are given. Simulations demonstrate that the corrected LR tests yield a realized size close to nominal level for both moderate *p* (around 20) and high dimension, while the traditional LR tests with *χ*^{2} approximation fails.

Another contribution from the paper is that for testing the equality between two covariance matrices, the proposed correction applies equally for non-Gaussian populations yielding a valid pseudo-likelihood ratio test.

#### Article information

**Source**

Ann. Statist., Volume 37, Number 6B (2009), 3822-3840.

**Dates**

First available in Project Euclid: 23 October 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1256303528

**Digital Object Identifier**

doi:10.1214/09-AOS694

**Mathematical Reviews number (MathSciNet)**

MR2572444

**Zentralblatt MATH identifier**

1360.62286

**Subjects**

Primary: 62H15: Hypothesis testing

Secondary: 62H10: Distribution of statistics

**Keywords**

High-dimensional data testing on covariance matrices Marčenko–Pastur distributions random F-matrices

#### Citation

Bai, Zhidong; Jiang, Dandan; Yao, Jian-Feng; Zheng, Shurong. Corrections to LRT on large-dimensional covariance matrix by RMT. Ann. Statist. 37 (2009), no. 6B, 3822--3840. doi:10.1214/09-AOS694. https://projecteuclid.org/euclid.aos/1256303528