# E-Statistics, Group Invariance and Anytime Valid Testing

arXiv: https://arxiv.org/abs/2208.07610 · HTML: https://mlsanity.com/paper/2208.07610/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Aug 2022.

## Built on by (most cited first, 8 of 8)

- [Protected Test-Time Adaptation via Online Entropy Matching: A Betting Approach](https://mlsanity.com/paper/2408.07511/) (2024)
- [Peeking with PEAK: Sequential, Nonparametric Composite Hypothesis Tests for Means of Multiple Data Streams](https://mlsanity.com/paper/2402.06122/) (2024)
- [The extended Ville's inequality for nonintegrable nonnegative supermartingales](https://mlsanity.com/paper/2304.01163/) (2023)
- [Combining Evidence Across Filtrations](https://mlsanity.com/paper/2402.09698/) (2024)
- [Fuzzy Prediction Sets: Conformal Prediction with E-values](https://mlsanity.com/paper/2509.13130/) (2025)
- [Anytime-valid t-tests and confidence sequences for Gaussian means with unknown variance](https://mlsanity.com/paper/2310.03722/) (2023)
- [Optimal Anytime-Valid Tests for Composite Nulls](https://mlsanity.com/paper/2512.20039/) (2025)
- [NxN E-valuation: Hypothesis Certification via a Conformal CRT Null](https://mlsanity.com/paper/2608.06621/) (2026)

## Abstract (arXiv)

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.