On Empirical Likelihood in Semiparametric Two-Sample Density Ratio Models

We consider estimation and test problems for some semiparametric two-sample density ratio models. The profile empirical likelihood (EL) poses an irregularity problem under the null hypothesis that the laws of the two samples are equal. We show that a “dual” form of the profile EL is well defined even under the null hypothesis. A statistical test, based on the dual form of the EL ratio statistic (ELRS), is then proposed. We give an interpretation for the dual form of the ELRS through φ-divergences and “duality” technique. The asymptotic properties of the test statistic are presented both under the null and the alternative hypotheses, and an approximation to the power function is deduced.