The Annals of Statistics

Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions

Tiefeng Jiang and Fan Yang

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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.

Article information

Ann. Statist., Volume 41, Number 4 (2013), 2029-2074.

First available in Project Euclid: 23 October 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Likelihood ratio test central limit theorem high-dimensional data multivariate normal distribution hypothesis test covariance matrix mean vector multivariate Gamma function


Jiang, Tiefeng; Yang, Fan. Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions. Ann. Statist. 41 (2013), no. 4, 2029--2074. doi:10.1214/13-AOS1134.

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