A Generalized Empirical Likelihood Approach for Two-Group Comparisons Given a U-Statistic Constraint

We investigate a generalized empirical likelihood approach in a two-group setting where the constraints on parameters have a form of U-statistics. In this situation, the summands that consist of the constraints for the empirical likelihood are not independent, and a weight of each summand may not have a direct interpretation as a probability point mass, dissimilar to the common empirical likelihood constraints based on independent summands. We show that the resulting empirical likelihood ratio statistic has a weighted chi-squared distribution in the univariate case and a combination of weighted chi-squared distributions in the multivariate case. Through an extensive Monte-Carlo study, we show that the proposed methods applied for some well-known U-statistics have robust Type I error control under various underlying distributions including cases with a violation of exchangeability under null hypotheses. For the application, we employ the proposed methods to test hypotheses in crossover designs demonstrating an adaptability of the proposed methods in various hypothesis tests.