  Recent work on game-theoretic statistics and safe anytime-valid inference
(SAVI) provides new tools for statistical inference without assuming any
regularity conditions. In particular, the framework of universal inference
proposed by Wasserman, Ramdas, and Balakrishnan (2020) offers new solutions by
modifying the likelihood ratio test in a data-splitting scheme. In this paper,
we study the performance of the resulting split likelihood ratio test under
Gaussian mixture models, which are canonical examples for models in which
classical regularity conditions fail to hold. We first establish that under the
null hypothesis, the split likelihood ratio statistic is asymptotically normal
with increasing mean and variance. Moreover, contradicting the usual belief
that the flexibility of SAVI and universal methods comes at the price of a
significant loss of power, we are able to prove that universal inference
surprisingly achieves the same detection rate $(n^{-1}\log\log n)^{1/2}$ as the
classical likelihood ratio test.
