Corrections to LRT on Large Dimensional Covariance Matrix by RMT

Zhidong Bai (1, 2) , Dandan Jiang (2) , Jian-Feng Yao (3) , Shurong Zheng (1)

1 KLASMOE - Key Laboratory of Applied Statistics under Ministry of Education
2 DSAP - Department of Statistics and Applied Probability
3 IRMAR - Institut de Recherche Mathématique de Rennes

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 is large compared to the sample size. 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 khi-square 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.

Mots clés: High-dimensional data; Testing on covariance matrices; Marcenko-Pastrur distributions; Random F-matrices

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Zhidong Bai, Dandan Jiang, Jian-Feng Yao, Shurong Zheng. Corrections to LRT on Large Dimensional Covariance Matrix by RMT. Annals of Statistics, 2009, 37 (6B), pp.3822-3840. ⟨10.1214/09-AOS694⟩. ⟨hal-00358579⟩