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 strategies are guided by the derivation of a simple log-optimal betting strategy, and display 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：我们提出了一种任意有效的置换检验方法，其中数据集固定且排列按顺序逐次抽样，旨在通过减少抽样次数和提前终止来节省计算资源。核心技术进展是开发了新的检验鞅（初始值为一的非负鞅），用于针对特定备择假设检验可交换性。这些检验鞅通过新颖且简单的下注策略构建，这些策略智能地押注于置换检验统计量的相对秩次。下注策略由简单的对数最优下注策略推导而来，并在实践中展现出卓越的检验力。与Besag和Clifford的知名方法相比，我们的方法能在任意停止时间产生有效的e值或p值，并且采用特定停止规则时，在零假设和备择假设下均能实现计算增益，且不损失检验力。