## The Annals of Statistics

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

#### Abstract

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

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

Dates
First available in Project Euclid: 23 October 2013

https://projecteuclid.org/euclid.aos/1382547512

Digital Object Identifier
doi:10.1214/13-AOS1134

Mathematical Reviews number (MathSciNet)
MR3127857

Zentralblatt MATH identifier
1277.62149

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1382547512

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