A two-dimensional Cramér–von Mises test for the two-sample problem with dispersion alternatives

Abstract

We consider the two-sample problem with dispersion alternatives. Starting with a sufficient characterization of the alternative by means of two integrals we come up with a test which is based on the empirical counterparts of the integrals. Especially the critical region is now the inverse image of an infinite rectangle in ℝ². The common limit distribution of the empirical integrals is determined under the hypothesis of randomness and under a broad class of nonparametric local alternatives. In each case it turns out to be normal. It enables the construction of an asymptotic level-α test, which is consistent on the alternative. In addition we are able to make local power investigations. In our example the new test is superior to the classical Cramér–von Mises test.