  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
finite-dimensional linear regression.
