On estimation of mean squared errors of benchmarked empirical Bayes estimators

TL;DR

This paper analyzes the impact of benchmarking constraints on the mean squared error of empirical Bayes estimators in small area estimation, providing asymptotic results, unbiased estimators, and bootstrap methods.

Contribution

It derives the asymptotic increase in MSE due to benchmarking and proposes unbiased and bootstrap estimators for this MSE under the Fay-Herriot model.

Findings

Benchmarking increases MSE by O(m^{-1})

Proposes an asymptotically unbiased MSE estimator

Develops a bootstrap estimator for MSE comparison

Abstract

We consider benchmarked empirical Bayes (EB) estimators under the basic area-level model of Fay and Herriot while requiring the standard benchmarking constraint. In this paper we determine the excess mean squared error (MSE) from constraining the estimates through benchmarking. We show that the increase due to benchmarking is O(m^{-1}), where m is the number of small areas. Furthermore, we find an asymptotically unbiased estimator of this MSE and compare it to the second-order approximation of the MSE of the EB estimator or, equivalently, of the MSE of the empirical best linear unbiased predictor (EBLUP), that was derived by Prasad and Rao (1990). Morever, using methods similar to those of Butar and Lahiri (2003), we compute a parametric bootstrap estimator of the MSE of the benchmarked EB estimator under the Fay-Herriot model and compare it to the MSE of the benchmarked EB estimator found by a second-order approximation. Finally, we illustrate our methods using SAIPE data from the U.S. Census Bureau, and in a simulation study.