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

TL;DR

This paper develops a generalized empirical likelihood method for two-group comparisons involving U-statistics, addressing dependence issues and providing robust hypothesis testing tools adaptable to various designs.

Contribution

It introduces a novel empirical likelihood approach for U-statistics constraints, deriving its distribution and demonstrating robustness in hypothesis testing.

Findings

Empirical likelihood ratio follows a weighted chi-squared distribution in univariate cases.

Method maintains robust Type I error control across diverse distributions.

Applicable to crossover designs and various hypothesis tests.

Abstract

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.