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 conditional $e$-values. To construct $e$-values, we first develop an essentially complete class of admissible $e$-values in which one can flexibly `plug in' almost any desired test statistic. To make the $e$-values conditional, we explore the intersection between the concepts of conditional invariance and sequential invariance, and find that the appropriate conditional distribution can be captured by a compact subgroup. To find powerful $e$-values for given alternatives, 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. We apply these statistics 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 yields an improved version of a known $e$-value for testing sign-symmetry. Moreover, we introduce an impatience parameter that allows users to obtain more power now in exchange for less power in the long run.

翻译：我们开发了一种灵活的在线版置换检验方法。该方法允许在数据到达时检验可交换性，且可在不破坏检验规模的条件下自由选择停止或继续。我们的方法可超越可交换性，推广至紧群作用下的其他形式的不变性。该方法的核心在于构建一个由多个条件$e$-值的连乘构成的$e$-过程。为构造$e$-值，我们首先建立了一个本质上完备的可容许$e$-值类，可灵活地“插入”几乎任意所需检验统计量。为实现条件$e$-值的构造，我们探究了条件不变性与序贯不变性概念之间的交集，发现恰当的条件分布可通过一个紧子群来刻画。为在给定备择假设下寻找有效的$e$-值，我们发展了检验群不变性的似然比理论，得到了群不变性检验的最优性新结论。这些统计量依备择假设的指定空间不同而呈现三种形态。我们将这些统计量应用于高斯位置偏移检验，得到了与球性检验中$t$-检验的关联、与检验可交换性时softmax函数及其温度参数的关联，并改进了已知的检验符号对称性的$e$-值。此外，我们引入了一个“耐心参数”，允许用户以长期检验功效为代价换取当前更高的检验功效。