Estimation of design‐based mean squared error of a small area mean model‐based estimator under a nested error linear regression model

Abstract

AbstractIn this article, we propose a conditional model estimator (cmmse) for the design‐based mean squared error (dMSE) of a small area mean estimator under the basic unit level model. The mean squared error dMSE refers to the variability of a small area estimator over all possible sample selections. It is different from the model mean squared error (mMSE), traditionally used to measure the efficiency in small area estimation problems. For known model parameters, Rao, Rubin‐Bleuer & Estevao [Rao et al., Survey Methodology 2018; 44, 151–166] showed that dMSE depends on two quadratic finite population parameters. A design estimator of dMSE, denoted as dmse, is obtained by substituting the quadratic parameters with their corresponding design unbiased estimators. Rao, Rubin‐Bleuer & Estevao [Rao et al., Survey Methodology 2018; 44, 151–166] proposed a composite MSE estimator (cmse) based on both the design and the model. This estimator is defined as a weighted average between the design‐based dmse and a model‐based estimator (mmse). Given known variance components, we obtain a new formula for dMSE that accounts for the estimation of the fixed model coefficients. Our conditional model MSE estimator cmmse is obtained by replacing the quadratic finite population parameters by their best predictions under the model, in the new formula of dMSE. Properties of the proposed estimator are studied in terms of design bias, relative root mean squared error, coverage rate and a score function of the confidence intervals.