E-Statistics, Group Invariance and Anytime Valid Testing
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We study worst-case growth-rate optimal (GROW) E-variables for hypothesis testing between two group models. If the underlying group G acts freely on the observation space, there exists a maximally invariant statistic of the data. We show that among all E-statistics, invariant or not, the likelihood ratio of the maximally invariant is GROW and that an anytime valid test can be based on this likelihood ratio. By virtue of a representation theorem of Wijsman, it is equivalent to a Bayes factor with a right Haar prior on G. Such Bayes factors are known to have good frequentist and Bayesian properties. We show that reductions through sufficiency and invariance can be made in tandem without affecting optimality. A crucial assumption on the group G is its amenability, a well-known group-theoretical condition, which holds for general scale- and location families as well as finite-dimensional linear regression.
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