  We propose a general method for constructing hypothesis tests and confidence
sets that have finite sample guarantees without regularity conditions. We refer
to such procedures as "universal." The method is very simple and is based on a
modified version of the usual likelihood ratio statistic, that we call "the
split likelihood ratio test" (split LRT). The method is especially appealing
for irregular statistical models. Canonical examples include mixture models and
models that arise in shape-constrained inference. Constructing tests and
confidence sets for such models is notoriously difficult. Typical inference
methods, like the likelihood ratio test, are not useful in these cases because
they have intractable limiting distributions. In contrast, the method we
suggest works for any parametric model and also for some nonparametric models.
The split LRT can also be used with profile likelihoods to deal with nuisance
parameters, and it can also be run sequentially to yield anytime-valid
$p$-values and confidence sequences.
