Anytime-Valid Tests of Group Invariance through Conformal Prediction

Many standard statistical hypothesis tests, including those for normality and exchangeability, can be reformulated as tests of invariance under a group of transformations. We develop anytime-valid tests of invariance under the action of general compact groups and show their optimality -- in a specific logarithmic-growth sense -- against certain alternatives. This is achieved by using the invariant structure of the problem to construct conformal test martingales, a class of objects associated to conformal prediction. We apply our methods to extend recent anytime-valid tests of independence, which leverage exchangeability, to work under general group invariances. Additionally, we show applications to testing for invariance under subgroups of rotations, which corresponds to testing the Gaussian-error assumptions behind linear models.