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 finite-dimensional linear regression.

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techReport E-statistics, group invariance and anytime valid testing M.F. Pérez (Muriel), T.D. Lardy (Tyron), R. de Heide (Rianne) and P.D. Grünwald (Peter)