Gaussian universal likelihood ratio testing.

The article focuses on the universal likelihood ratio test (universal LRT), a recent hypothesis testing framework that is valid in finite samples without requiring regularity conditions, contrasting it with the classical likelihood ratio test (LRT) in the setting of testing the mean of multivariate Gaussian data. It analyzes several variants of the universal LRT—including split, cross-fit, and repeated subsampling approaches—demonstrating that repeated subsampling yields the best balance of size and power, with confidence sets approximately 1.5 times larger in squared radius than classical LRT sets in high dimensions. The work also presents an example testing a nonconvex “doughnut-shaped” null hypothesis, showing that universal LRT methods can achieve higher power than classical approaches in such complex settings. These findings suggest that universal LRT provides a flexible and valid testing framework applicable to problems where classical methods fail or are intractable, including mixture model component testing and shape-constrained density estimation.