E-statistics, group invariance and anytime-valid testing

We study worst-case-growth-rate-optimal (GROW) e-statistics for hypothesis
testing between two group models. It is known that under a mild
condition on the action of the underlying group G on the data, there exists
a maximally invariant statistic. We show that among all e-statistics, invariant
or not, the likelihood ratio of the maximally invariant statistic is GROW,
both in the absolute and in the relative sense, and that an anytime-valid test
can be based on it. The GROW e-statistic is equal to a Bayes factor with a
right Haar prior on G. Our treatment avoids nonuniqueness issues that sometimes
arise for such priors in Bayesian contexts. A crucial assumption on the
group G is its amenability, a well-known group-theoretical condition, which
holds, for instance, in scale-location families. Our results also apply to finitedimensional
linear regression.