Matching posterior and frequentist cumulative distribution functions with empirical-type likelihoods in the multiparameter case

We consider a general class of empirical-type likelihoods for a vector-valued population mean. Members of the class that admit a matching prior, in a higher order asymptotic sense, for the posterior and frequentist cumulative distribution functions are characterized. The results are compared and contrasted with those on posterior quantiles in the scalar-valued case.