Hash-augmented adaptive multilevel splitting Monte Carlo algorithm for accurate estimation of two-sample permutation test p-values
Nonparametric permutation tests are widely used for statistical analysis. However, exact computation of test p-values can be algorithmically challenging, particularly for custom tests with complex test statistics. In contrast, Monte Carlo sampling can be easily applied to any test statistic, but it suffers from poor relative accuracy when estimating small p-values, interfering with multiple hypothesis testing correction and leading to other issues. In this work, we present a hash-augmented adaptive multilevel splitting Monte Carlo algorithm that enables accurate estimation of arbitrarily small p-values in two-sample permutation tests. Using the Kolmogorov-Smirnov and the Mann-Whitney U tests as examples, we highlight potential pitfalls related to the discreteness of the test statistic distribution and show how to address them. By comparing with an exact algorithm, we demonstrate the accuracy of the p-value estimates provided by the proposed algorithm and the validity of the associated confidence intervals. We provide a reference implementation of the proposed algorithm in the Python package hamstest, which allows p-value estimation for a user-defined statistic.