A two-sample nonparametric test for one-sided location-scale alternative

An increase in location is often accompanied by an increase in variability. Moreover, in randomized studies, the presence of heteroscedasticity can indicate a treatment effect. In these cases a location-scale test is appropriate. A common approach of a location-scale test is to use the sum of a location and a scale test statistic. In this study, we focus on one-sided location-scale tests. We introduce a one-sided Lepage-type test statistic. Since right-skewed data are common in many real-world applications, we design test statistics that are powerful also for right-skewed data. Moreover, this paper introduces both a maximum test and an adaptive test utilizing the one-sided Lepage-type test statistics. The limiting distributions of the maximum test statistics are also derived. We assess the performance of the different test statistics in various scenarios for continuous distributions through Monte Carlo simulations. Our simulation results demonstrate that the proposed new test statistics can significantly increase and stabilize statistical power, making it strong competitors to existing location-scale tests. We present practical illustrations based on example data and conclude with some final remarks.