Central Limit Theorems for Classical Likelihood Ratio Tests for High-Dimensional Normal Distributions

TL;DR

This paper establishes central limit theorems for likelihood ratio tests in high-dimensional normal distributions, showing they converge to normal distributions and outperform traditional chi-square approximations in such settings.

Contribution

It provides the first CLTs for LRTs in high-dimensional normal models where both p and n grow proportionally, with explicit means and variances.

Findings

Likelihood ratio test statistics converge to normal distributions in high dimensions.

Simulations show improved accuracy over chi-square approximations.

The results facilitate more reliable inference in high-dimensional multivariate analysis.

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/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.