EXACT AND ASYMPTOTICALLY ROBUST PERMUTATION TESTS

Given independent samples from P and Q, two-sample permutation tests allow one to construct exact level tests when the null hypothesis is P = Q. On the other hand, when comparing or testing particular parameters . of P and Q, such as their means or medians, permutation tests need not be level ., or even approximately level . in large samples. Under very weak assumptions for comparing estimators, we provide a general test procedure whereby the asymptotic validity of the permutation test holds while retaining the exact rejection probability . in finite samples when the underlying distributions are identical. The ideas are broadly applicable and special attention is given to the k-sample problem of comparing general parameters, whereby a permutation test is constructed which is exact level . under the hypothesis of identical distributions, but has asymptotic rejection probability . under the more general null hypothesis of equality of parameters. A Monte Carlo simulation study is performed as well. A quite general theory is possible based on a coupling construction, as well as a key contiguity argument for the multinomial and multivariate hypergeometric distributions.