E-statistics, group invariance and any time valid testing

Muriel Felipe Pérez-Ortiz, Tyron Lardy, Rianne de Heide and Peter Grünwald

What is this paper about?

This paper connects e-values, symmetry and group invariance. For testing between invariant models it identifies a growth-rate-optimal e-statistic through a maximally invariant statistic, and relates it to Bayes factors based on right Haar priors.

Summary

We study worst-case-growth-rate-optimal (GROW) e-statistics for hypothesis testing between two group models. Under a mild condition on the action of the underlying group on the data, there exists a maximally invariant statistic. We show that, among all e-statistics, invariant or not, the likelihood ratio of this maximally invariant statistic is GROW in both the absolute and relative sense, and that an anytime-valid test can be based on it. The GROW e-statistic equals a Bayes factor with a right Haar prior on the group. A crucial assumption is amenability of the group, which holds for example in scale-location families. The results also apply to finite-dimensional linear regression.

Topics

E-values and anytime-valid inference · Foundations of statistics, probability and learning

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