Abstract: This study examines a measure of uncertainty for robust small area estimation (SAE). We consider the estimation of the mean squared prediction error (MSPE) of the observed best predictor (OBP) in SAE under the Fay-Herriot model with potential model misspecification. Previously, it was thought that the traditional Prasad-Rao (PR) linearization method could not be used, because it is derived under the assumption that the underlying model is correctly specified. However, we show that when it comes to estimating the unconditional MSPE, the PR estimator, derived for estimating the MSPE of the OBP, assuming that the underlying model is correct, remains first-order unbiased, even when the underlying model is misspecified in its mean function. A second-order unbiased estimator of the MSPE is derived by modifying the PR MSPE estimator. The PR and modified PR estimators also have much smaller variation than that of existing MSPE estimators for the OBP. The theoretical findings are supported by empirical results, including simulation studies and real-data applications.

Key words and phrases: Fay-Herriot model, model misspecification, observed best prediction, robustness, second-order unbiasedness, small area estimation.