The Annals of Statistics

Corrections to LRT on large-dimensional covariance matrix by RMT

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

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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 χ2 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

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

First available in Project Euclid: 23 October 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H15: Hypothesis testing
Secondary: 62H10: Distribution of statistics

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


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.

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