E-Statistics, Group Invariance and Anytime Valid Testing

arXiv (Cornell University)

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.

Authors 4

1. Muriel Felipe Pérez-Ortiz

   Eindhoven University of Technology

   Affiliation as printed

   Eindhoven University of Technology , Eindhoven , The Netherlands ,

2. Tyron Lardy Aachen

   Leiden University

   Affiliation as printed

   Leiden University , Leiden , The Netherlands ,

3. Rianne de Heide

   Vrije Universiteit Amsterdam

   Affiliation as printed

   Vrije Universiteit , Amsterdam , The Netherlands ,

4. Peter Grünwald

   Centrum Wiskunde & Informatica

   Affiliation as printed

   Centrum Wiskunde & Informatica , Amsterdam , The Netherlands ,