Corrections to LRT on Large Dimensional Covariance Matrix by RMT

Zhidong Bai; Dandan Jiang; Jian-Feng Yao; Shurong Zheng

hal-00358579, version 1

Corrections to LRT on Large Dimensional Covariance Matrix by RMT

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

The Annals of Statistics 37, 6B (2009) 3822-3840

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 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.

1: Key Laboratory of Applied Statistics under Ministry of Education (KLASMOE)
Northeast Normal University
2: Department of Statistics and Applied Probability (DSAP)
National University of Singapore
3: Institut de Recherche Mathématique de Rennes (IRMAR)
CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne

hal-00358579, version 1
http://hal.archives-ouvertes.fr/hal-00358579
oai:hal.archives-ouvertes.fr:hal-00358579
From: Jian-Feng Yao <>
Submitted on: Tuesday, 3 February 2009 17:23:56
Updated on: Friday, 19 March 2010 10:19:32

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DOI: 10.1214/09-AOS694

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%% hal-00358579, version 1
%% http://hal.archives-ouvertes.fr/hal-00358579
@article{bai:hal-00358579,
    hal_id = {hal-00358579},
    url = {http://hal.archives-ouvertes.fr/hal-00358579},
    title = {{Corrections to LRT on Large Dimensional Covariance Matrix by RMT}},
    author = {Bai, Zhidong and Jiang, Dandan and Yao, Jian-Feng and Zheng, Shurong},
    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 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.}},
    keywords = {High-dimensional data ; Testing on covariance matrices ; Marcenko-Pastrur distributions ; Random F-matrices},
    language = {English},
    affiliation = {Key Laboratory of Applied Statistics under Ministry of Education - KLASMOE , Department of Statistics and Applied Probability - DSAP , Institut de Recherche Math{\'e}matique de Rennes - IRMAR},
    pages = {3822-3840},
    journal = {The Annals of Statistics},
    volume = {37},
    number = {6B },
    note = {26 pages, 2 figures and 3 tables. CNSF grant 10571020 ; NUS grant R-155-000-061-112 },
    audience = {international },
    doi = {10.1214/09-AOS694 },
    year = {2009},
    pdf = {http://hal.archives-ouvertes.fr/hal-00358579/PDF/st-long.pdf},
}

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