Bootstrap Consistency for Empirical Likelihood in Density Ratio Models

We establish the validity of bootstrap methods for empirical likelihood (EL) inference under the density ratio model (DRM).In particular, we prove that the bootstrap maximum EL estimators share the same limiting distribution as their population counterparts, both at the parameter level and for distribution functionals.Our results extend existing pointwise convergence theory to weak convergence of processes, which in turn justifies bootstrap inference for quantiles and dominance indices within the DRM framework.These theoretical guarantees close an important gap in the literature, providing rigorous foundations for resampling-based confidence intervals and hypothesis tests.Simulation studies further demonstrate the accuracy and practical value of the proposed approach.