  The classical likelihood ratio test (LRT) based on the asymptotic chi-squared
distribution of the log likelihood is one of the fundamental tools of
statistical inference. A recent universal LRT approach based on sample
splitting provides valid hypothesis tests and confidence sets in any setting
for which we can compute the split likelihood ratio statistic (or, more
generally, an upper bound on the null maximum likelihood). The universal LRT is
valid in finite samples and without regularity conditions. This test empowers
statisticians to construct tests in settings for which no valid hypothesis test
previously existed. For the simple but fundamental case of testing the
population mean of d-dimensional Gaussian data with identity covariance matrix,
the classical LRT itself applies. Thus, this setting serves as a perfect test
bed to compare the classical LRT against the universal LRT. This work presents
the first in-depth exploration of the size, power, and relationships between
several universal LRT variants. We show that a repeated subsampling approach is
the best choice in terms of size and power. For large numbers of subsamples,
the repeated subsampling set is approximately spherical. We observe reasonable
performance even in a high-dimensional setting, where the expected squared
radius of the best universal LRT's confidence set is approximately 3/2 times
the squared radius of the classical LRT's spherical confidence set. We
illustrate the benefits of the universal LRT through testing a non-convex
doughnut-shaped null hypothesis, where a universal inference procedure can have
higher power than a standard approach.
