Data-dependent probability matching priors for highest posterior density and equal-tailed two-sided regions based on empirical-type likelihoods

We consider a very general class of empirical-type likelihoods which includes the usual empirical likelihood and all its major variants proposed in the literature. It is known that none of these likelihoods admits a data-free probability matching prior for the highest posterior density region. We develop necessary higher order asymptotics to show that at least for the usual empirical likelihood this difficulty can be resolved if data-dependent priors are entertained. A related problem concerning the equal-tailed two-sided posterior credible region is also investigated. A simulation study is seen to lend support to the theoretical results.