Asymptotic power of likelihood ratio tests for high dimensional data

TL;DR

This paper analyzes the asymptotic power of likelihood ratio tests for identity hypotheses in high-dimensional settings, deriving explicit power expressions and demonstrating LRT's effectiveness in detecting small eigenvalues.

Contribution

It provides the first explicit asymptotic power formula for LRT in high-dimensional identity testing, supported by simulation comparisons.

Findings

LRT has high power to detect eigenvalues near zero in high dimensions

Explicit asymptotic power expressions are derived for the LRT

Simulation results show LRT outperforms other tests in certain scenarios

Abstract

This paper considers the asymptotic power of likelihood ratio test (LRT) for the identity test when the dimension p is large compared to the sample size n. The asymptotic distribution of LRT under alternatives is given and an explicit expression of the power is derived. A simulation study is carried out to compare LRT with other tests. All these studies show that LRT is a powerful test to detect eigenvalues around zero. Key words and phrases: Covariance matrix, High dimensional data, Identity test, Likelihood ratio test, Power