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.

翻译：我们提出了一种灵活的在线置换检验方法。该方法允许在数据流式到达时检验可交换性，且可在不破坏检验尺度有效性的前提下自主决定终止或继续观测。我们的方法突破了可交换性假设的局限，适用于紧致群作用下的其他不变性形式。该技术通过构建由批观测数据生成的多个$e$-值的连乘积形式的$e$-过程来实现。在$e$-值构建方面，我们首先发展了一类本质完备的可容许$e$-值族，可灵活'嵌入'任意所需检验统计量。为获取优质$e$-值，我们建立了群不变性检验的似然比理论，得到了群不变性检验的最优性新结果。根据备择假设设定空间的不同，这些统计量呈现三种不同形态，其诱导的$e$-过程具备良好的势性质。我们将所提统计量应用于高斯位置偏移检验：检验球对称性时与$t$-检验建立联系；检验可交换性时与softmax函数及其温度参数产生关联；检验符号对称性时则获得适用于任意相依结构的$e$-过程。