E-statistics, group invariance and anytime-valid testing
Muriel Felipe Pérez-Ortiz, Tyron Lardy, Rianne de Heide, Peter D. Grünwald
The Annals of Statistics · 2024 · Volume 52 · Issue 4
Abstract
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.
Keywords
62B99, 62F03, 62L10, anytime-valid methods, Bayes factors, e-values, group invariance, right Haar priors
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