A Paradox about Likelihood Ratios?

TL;DR

This paper investigates whether the asymptotic distribution of the log-likelihood ratio test statistic is Gaussian or chi-squared, using two simple examples to shed light on this distinction.

Contribution

It provides new insights into the conditions under which the log-likelihood ratio test statistic follows Gaussian or chi-squared distributions.

Findings

Examples illustrate the divergence from classical assumptions

Clarifies when Gaussian vs. chi-squared distributions apply

Highlights potential paradox in likelihood ratio testing

Abstract

We consider whether the asymptotic distributions for the log-likelihood ratio test statistic are expected to be Gaussian or chi-squared. Two straightforward examples provide insight on the difference.