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August 2013 Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions
Tiefeng Jiang, Fan Yang
Ann. Statist. 41(4): 2029-2074 (August 2013). DOI: 10.1214/13-AOS1134


For random samples of size $n$ obtained from $p$-variate normal distributions, we consider the classical likelihood ratio tests (LRT) for their means and covariance matrices in the high-dimensional setting. These test statistics have been extensively studied in multivariate analysis, and their limiting distributions under the null hypothesis were proved to be chi-square distributions as $n$ goes to infinity and $p$ remains fixed. In this paper, we consider the high-dimensional case where both $p$ and $n$ go to infinity with $p/n\to y\in(0,1]$. We prove that the likelihood ratio test statistics under this assumption will converge in distribution to normal distributions with explicit means and variances. We perform the simulation study to show that the likelihood ratio tests using our central limit theorems outperform those using the traditional chi-square approximations for analyzing high-dimensional data.


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Tiefeng Jiang. Fan Yang. "Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions." Ann. Statist. 41 (4) 2029 - 2074, August 2013.


Published: August 2013
First available in Project Euclid: 23 October 2013

zbMATH: 1277.62149
MathSciNet: MR3127857
Digital Object Identifier: 10.1214/13-AOS1134

Primary: 62H15
Secondary: 62H10

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 4 • August 2013
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