An adaptive distribution-free test for the general two-sample problem

Summary For the general two-sample problem we propose an adaptive test which is based on tests of Kolmogorov-Smirnov- and Cramèr- von Mises type. These tests are modifications of the Kolmogorov-Smirnov- and Cramèr- von Mises tests by using various weight functions in order to obtain higher power than its classical counterparts for short-tailed and right-skewed distributions. In practice, however, we generally have no information about the underlying distribution of the data. Thus, an adaptive test should be applied which takes into account the given data. The proposed adaptive test is based on Hogg’s concept, i.e., first, to classify the unknown distribution function with respect to two measures, one for skewness and one for tailweight, and second, to use an appropriate test of Kolmogorov-Smimov- and Cramèr- von Mises type for this classified type of distribution. We compare the distribution-free adaptive test with the tests of Kolmogorov-Smirnov- and Cramèr-von Mises type as well as with the Lepage test and a modification of its in the case of location and scale alternatives including the same shape and different shapes of the distributions of the X- and Y- variables. The power comparison of the tests is carried out via Monte Carlo simulation assuming short-, medium- and long-tailed distributions as well as distributions skewed to the right. It turns out that, on the whole, the adaptive test is the best one for the broad class of distributions considered.