  Benchmarking estimation and its risk evaluation is a practically important
issue in small area estimation. While hierarchical Bayesian methods have been
widely adopted in small area estimation, a unified Bayesian approach to
benchmarking estimation has not been fully discussed. This work employs an
entropic tilting method to modify the posterior distribution of the small area
parameters to meet the benchmarking constraint, which enables us to obtain
benchmarked point estimation as well as reasonable uncertainty quantification.
Using conditionally independent structures of the posterior, we first introduce
general Monte Carlo methods for obtaining a benchmarked posterior and then show
that the benchmarked posterior can be obtained in an analytical form for some
representative small area models. We demonstrate the usefulness of the proposed
method through simulation and empirical studies.
