Abstract
Several methods for comparing k populations have been proposed in the literature. These methods assess the same null hypothesis of equal distributions but differ in the alternative hypothesis they consider. We focus on two important alternative hypotheses: monotone and umbrella ordering. Two new families of test statistics are proposed, including two known tests, as well as two new powerful tests under monotone ordering. Furthermore, these families are adapted for testing umbrella ordering. We compare some members of the families with respect to power and Type I errors under different simulation scenarios. Finally, the methods are illustrated in several applications to real data.