This paper studies decision theoretic properties of benchmarked estimators which
are of some importance in small area estimation problems. Benchmarking is intended
to improve certain aggregate properties (such as study-wide averages) when
model based estimates have been applied to individual small areas. We study admissibility
and minimaxity properties of such estimators by reducing the problem
to one of studying these problems in a related derived problem. For certain such
problems we show that unconstrained solutions in the original (unbenchmarked)
problem give unconstrained Bayes, minimax or admissible estimators which automatically
satisfy the benchmark constraint. We illustrate the results with several
examples. Also, minimaxity of a benchmarked empirical Bayes estimator is shown
in the Fay-Herriot model, a frequently used model in small area estimation.
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