EStatistics, Group Invariance and Anytime Valid Testing
Abstract
We study worstcase growthrate optimal (GROW) Evariables 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 Estatistics, 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 wellknown grouptheoretical condition, which holds for general scale and location families as well as finitedimensional linear regression.
 Publication:

arXiv eprints
 Pub Date:
 August 2022
 DOI:
 10.48550/arXiv.2208.07610
 arXiv:
 arXiv:2208.07610
 Bibcode:
 2022arXiv220807610P
 Keywords:

 Mathematics  Statistics Theory;
 Statistics  Methodology