An adaptive distribution-free test for the general two-sample problem (Q1855641)

The author considers the two-sample problem. Let \(x_1,\ldots,x_{m}\) and \(y_1,\ldots,y_{n}\) be independent random variables with absolutely continuous distribution functions \(F\) and \(G\), respectively. We wish to test the hypothesis \(H_0: G(z)=F(z)\) for all \(z\in R\) versus \(H_{1}: G(z)\neq F(z)\) for at least one \(z\in R\). As a special case the location-scale alternative is considered. Modifications of Kolmogorov-Smirnov and Cramér-von Mises tests by using appropriate weight functions in order to obtain higher power than their classical counterparts for short-tailed distributions and distributions skewed to the right are presented. For adaptive tests the author proposes first to classify the unknown distribution function with respect to two measures, one for skewness and one for tail-weight, and secondly to apply an appropriate test of Kolmogorov-Smirnov and Cramér-von Mises type. The power comparisons of the tests are carried out via Monte Carlo simulations not only for location and scale alternatives, but also for general shape alternatives.