Asymptotics and e ectiveness of conditional tests with applications to randomization tests

Conditional tests like two-sample permutation tests establish valid nite sample tests for large classes of nonparametric composite null hypotheses. In particular, we discuss the quality of various conditional distribution free tests in comparison to a (parametric) benchmark test. It is shown that they are often asymptotically e ective. This means that conditional tests are asymptotically equivalent to benchmark tests and do not loose asymptotic power under contiguous alternatives. Moreover they turn out to be asymptotically e cient for various models. The results apply to permutation tests, symmetry tests and tests based on U -statistics or maximum likelihood estimators. A counterexample for the Bootstrap shows that the principle is not always applicable for arbitrary resampling tests.