Robust Estimation of Mean Squared Error of Small Area Estimators

Abstract

A well-known model, due to Fay and Herriot, for estimating small area (domain) means, μ i , is considered. Given μ i 's, it is assumed that the survey estimators, y i , are independent with means μ i and known variances Di, i = 1, …, t. Further, the μ i 's are assumed to be independent with means x′iβ and unknown variance A, where x i is a vector of benchmark variables related to μ i and β is a vector of regression parameters. An empirical best linear unbiased prediction (EBLUP) estimator or an empirical linear Bayes estimator, t i (Â, y), of μ i is obtained. It is shown that an estimator of mean squared error (MSE) of t t (Â, y), derived by Prasad and Rao under normality of μ i and y i given μ i , is robust with respect to nonnormality of the μ i 's. Specifically, it is shown that the Prasad–Rao estimator of MSE is correct to terms of order O(t −1) for large t, assuming only certain moment conditions on the μ i 's and normally distributed survey errors. Results of a simulation study on the accuracy of the estimator of MSE, under nonnormality of the μ i 's, are also presented.