E-Statistics, Group Invariance and Anytime Valid Testing

TL;DR

This paper introduces GROW e-statistics for hypothesis testing in group models, showing that the likelihood ratio of the maximally invariant statistic is optimal and can be used for anytime-valid tests, with broad applicability including linear regression.

Contribution

It establishes the optimality of the likelihood ratio of the maximally invariant statistic as a GROW e-statistic and connects it to Bayes factors with Haar priors, avoiding nonuniqueness issues.

Findings

Likelihood ratio of maximally invariant statistic is GROW.

GROW e-statistic equals a Bayes factor with Haar prior.

Results apply to scale-location families and linear regression.

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.