  We develop constrained Bayesian estimation methods for small area problems:
those requiring smoothness with respect to similarity across areas, such as
geographic proximity or clustering by covariates; and benchmarking constraints,
requiring (weighted) means of estimates to agree across levels of aggregation.
We develop methods for constrained estimation decision-theoretically and
discuss their geometric interpretation. Our constrained estimators are the
solutions to tractable optimization problems and have closed-form solutions.
Mean squared errors of the constrained estimators are calculated via
bootstrapping. Our techniques are free of distributional assumptions and apply
whether the estimator is linear or non-linear, univariate or multivariate. We
illustrate our methods using data from the U.S. Census's Small Area Income and
Poverty Estimates program.
