Comparison of empirical likelihood and its dual likelihood under density ratio model

Density ratio models (DRMs) are commonly used semiparametric models to link related populations. Empirical likelihood (EL) under DRM has been demonstrated to be a flexible and useful platform for semiparametric inferences. Since DRM-based EL has the same maximum point and maximum likelihood as its dual form (dual EL), EL-based inferences under DRM are usually made through the latter. A natural question comes up: is there any efficiency loss of doing so? We make a careful comparison of the dual EL and DRM-based EL estimation methods from theory and numerical simulations. We find that their point estimators for any parameter are exactly the same, while they may have different performances in interval estimation. In terms of coverage accuracy, the two intervals are comparable for non- or moderate skewed populations, and the DRM-based EL interval can be much superior for severely skewed populations. A real data example is analysed for illustration purpose.