The assumption that data are invariant under the action of a compact group is implicit in many statistical modeling assumptions such as normality, or the assumption of independence and identical distributions. Hence, testing for the presence of such invariances offers a principled way to falsify various statistical models. In this article, we develop sequential, anytime-valid tests of distributional symmetry under the action of general compact groups. The tests that are developed allow for the continuous monitoring of data as it is collected while keeping type-I error guarantees, and include tests for exchangeability and rotational symmetry as special examples. The main tool to this end is the machinery developed for conformal prediction. The resulting test statistic, called a conformal martingale, can be interpreted as a likelihood ratio. We use this interpretation to show that the test statistics are optimal -- in a specific log-optimality sense -- against certain alternatives. Furthermore, we draw a connection between conformal prediction, anytime-valid tests of distributional invariance, and current developments on anytime-valid testing. In particular, we extend existing anytime-valid tests of independence, which leverage exchangeability, to work under general group invariances. Additionally, we discuss testing for invariance under subgroups of the permutation group and orthogonal group, the latter of which corresponds to testing the assumptions behind linear regression models.

翻译：数据在紧群作用下具有不变性的假设是许多统计建模假设（如正态性、独立同分布假设）的隐含前提。因此，检验此类不变性的存在性，为证伪各类统计模型提供了原则性方法。本文针对一般紧群作用下分布对称性的序列化即时有效检验展开研究。所提出的检验方法可在数据收集过程中实现连续监测的同时控制第一类错误概率，并以交换性和旋转对称性检验为特例。实现该目标的核心工具是基于保形预测理论构建的框架。由此得到的检验统计量——保形鞅——可被解释为似然比。我们利用这一解释，证明该检验统计量在特定对数最优性意义下对某些备择假设具有最优性。进一步地，我们揭示了保形预测、分布不变性即时有效检验与当前即时有效检验研究进展之间的关联。特别是，我们将现有利用交换性的独立即时有效检验拓展至一般群不变性场景。此外，我们还讨论了置换群与正交群子群下的不变性检验——后者对应线性回归模型假设的验证。