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.

翻译：本文研究了两个群模型间假设检验中的最坏情况增长率最优（GROW）e-统计量。已知在数据上底层群G的作用满足一定温和条件时，存在一个最大不变统计量。我们证明，在所有e-统计量（无论是否具有不变性）中，最大不变统计量的似然比在绝对意义和相对意义上均为GROW，并且可基于该统计量构建任意时刻有效的检验。该GROW e-统计量等于以G上的右哈尔先验为基础的贝叶斯因子。我们的处理避免了贝叶斯背景下此类先验有时出现的非唯一性问题。关于群G的一个关键假设是其可均性（一种广为人知的群论条件），该条件在例如尺度-位置族中成立。我们的结果同样适用于有限维线性回归。