We develop an anytime-valid permutation test, where the dataset is fixed and the permutations are sampled sequentially one by one, with the objective of saving computational resources by sampling fewer permutations and stopping early. The core technical advance is the development of new test martingales (nonnegative martingales with initial value one) for testing exchangeability against a very particular alternative. These test martingales are constructed using new and simple betting strategies that smartly bet on the relative ranks of permuted test statistics. The betting strategy is guided by the derivation of a simple log-optimal betting strategy, and displays excellent power in practice. In contrast to a well-known method by Besag and Clifford, our method yields a valid e-value or a p-value at any stopping time, and with particular stopping rules, it yields computational gains under both the null and the alternative without compromising power.

翻译：我们开发了一种时序有效的排列检验方法，其中数据集固定且排列按序逐个采样，旨在通过减少采样排列次数并提前停止来节省计算资源。核心技术进步在于针对特定备择假设检验可交换性，构建了新的检验鞅（初值为1的非负鞅）。这些检验鞅通过新颖简单的打赌策略构建，该策略智能地押注于排列后检验统计量的相对秩次。打赌策略的制定得益于简单对数最优打赌策略的推导，在实践中展现出卓越的检验功效。与Besag和Clifford的经典方法相比，我们的方法能在任意停止时间生成有效的e值或p值，且在特定停止规则下，能在保持检验功效的同时，于原假设和备择假设下均实现计算效率的提升。