Sequential permutation testing by betting

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.