The size distortion of bootstrap tests

We provide a theoretical framework in which to study the accuracy of bootst rap P values, which may be based on a parametric or nonparametric bootstrap . In the parametric case, the accuracy of a bootstrap test will depend on t he shape of what we call the critical value function. We show that, in many circumstances, the error in rejection probability of a bootstrap test will be one whole order of magnitude smaller than that of the corresponding asy mptotic test. We also propose a simulation method for estimating this error that requires the calculation of only two test statistics per replication.