Online Permutation Tests and Likelihood Ratios for Testing Group Invariance

We develop a flexible online version of the permutation test. This allows us to test exchangeability as the data is arriving, where we can choose to stop or continue without invalidating the size of the test. Our methods generalize beyond exchangeability to other forms of invariance under a compact group. Our approach relies on constructing an $e$-process that is the running product of multiple $e$-values that are constructed on batches of observations. To construct $e$-values, we first develop an essentially complete class of admissible $e$-values in which one can flexibly `plug' almost any desired test statistic. To find good $e$-values, we develop the theory of likelihood ratios for testing group invariance yielding new optimality results for group invariance tests. These statistics turn out to exist in three different flavors, depending on the space on which we specify our alternative, and their induced $e$-processes satisfy attractive power properties. We apply these statistic to test against a Gaussian location shift, which yields connections to the $t$-test when testing sphericity, connections to the softmax function and its temperature when testing exchangeability, and an $e$-process that is valid under arbitrary dependence when testing sign-symmetry.