Maintaining the exchangeability assumption for a two-group permutation test in the non-randomized setting

In this note, we develop a new two-group bootstrap-permutation test that utilizes the tail-extrapolated quantile function estimator for the bootstrap component. This test is an extension of the standard twogroup permutation test, that through its construction is defined to meet the exchangeability assumption, and thus it guarantees that the type I error is appropriately bounded by definition. This methodology is particularly useful in the non-randomized two-group setting for which the exchangeability assumption for the traditional two-group permutation test is untestable. We develop some theoretical results for the new test, followed by a simulation study and an example.