On the Phase Transition of Wilk's Phenomenon

TL;DR

This paper investigates the conditions under which Wilk's theorem's chi-squared approximations hold in high-dimensional data, providing guidelines for their application in multivariate hypothesis testing.

Contribution

It establishes phase transition criteria for Wilk's phenomenon in high dimensions and analyzes the accuracy of chi-squared approximations through asymptotic bias derivations.

Findings

Identifies phase transition points for Wilk's phenomenon in high-dimensional settings.

Provides asymptotic bias formulas for chi-squared approximations.

Offers statistical guidelines for choosing between conventional and new approximations.

Abstract

Wilk's theorem, which offers universal chi-squared approximations for likelihood ratio tests, is widely used in many scientific hypothesis testing problems. For modern datasets with increasing dimension, researchers have found that the conventional Wilk's phenomenon of the likelihood ratio test statistic often fails. Although new approximations have been proposed in high dimensional settings, there still lacks a clear statistical guideline regarding how to choose between the conventional and newly proposed approximations, especially for moderate-dimensional data. To address this issue, we develop the necessary and sufficient phase transition conditions for Wilk's phenomenon under popular tests on multivariate mean and covariance structures. Moreover, we provide an in-depth analysis of the accuracy of chi-squared approximations by deriving their asymptotic biases. These results may provide helpful insights into the use of chi-squared approximations in scientific practices.