An adaptive two-sample location-scale test of Lepage type for symetric distributions

For the two-sample location and scale problem we propose an adaptive test which is based on so called Lepage type tests. The well known test of Lepage (1971) is a confirmation of the Wilcoxon test for location alternatives and the Ansari-Bradley test for scale alternatives and it behaves well for symmetric and medium-tailed distributions. For the case of short-, medium- and long-tailed distributions we replace the Wilcoxon test and the Ansari-Bradley test by suitable other two-sample tests for location and scale, respectively, in order to get higher power than the classical Lepage test for such distributions. These tests here are called Lepage type tests. In practice, however, we generally have no clear idea about the distribution having generated our data. Thus, an adaptive test should be applied which takes the given data set into consideration. The proposed adaptive test is based on the concept of Hogg (1974), i.e., first, to classify the unknown symmetric distribution function with respect to a measure for tailweight and second, to apply an appropriate Lepage type test for this classified type of distribution. We compare the adaptive test with the three Lepage type tests in the adaptive scheme and with the classical Lepage test as well as with other parametric and nonparametric tests. The power comparison is carried out via Monte Carlo simulation. It is shown that the adaptive test is the best one for the broad class of distributions considered.