Studentized permutation tests for non-i.i.d. hypotheses and the generalized Behrens-Fisher problem

Abstract

It is shown that permutation tests based on studentized statistics are asymptotically exact of size α also under certain extended non-i.i.d. null hypotheses. To demonstrate the principle the results are applied to the generalized two-sample Behrens-Fisher problem for testing equality of the means under general non-parametric heterogeneous error distributions. Within this setting we propose a permutation version of the Welch test which is an extension of Pitman's two-sample permutation test. These results are special cases of a conditional central limit theorem for studentized permutation statistics which also applies to asymptotic power functions.