A non-parametric test of homogeneity for umbrella ordering alternative based on an isotonic estimators

Although there are many methods to test of homogeneity under umbrella ordering alternatives, most of these tests are aimed at the case of multinomial distributions or parametric families of continuous distributions. In this paper, we focus on testing homogeneity for umbrella ordering alternatives under nonparametric continuous distributions, considering both scenarios in which the peak of the umbrella ordering is known and unknown. The test statistics are constructed using isotonic estimates of the distribution functions, and the asymptotic distributions of the test statistics are derived. Critical values are obtained via a bootstrap procedure, and simulation results are provided to illustrate the performance of the proposed testing method.