Measuring uncertainty associated with model-based small area estimators

Domains (or subpopulations) with small sample sizes are called small areas. Traditional direct estimators for small areas do not provide adequate precision because the area-specific sample sizes are small. On the other hand, demand for reliable small area statistics has greatly increased. Model-based indirect estimators of small area means or totals are currently used to address difficulties with direct estimation. These estimators are based on linking models that borrow information across areas to increase the efficiency. In particular, empirical best (EB) estimators under area level and unit level linear regression models with random small area effects have received a lot of attention in the literature. Model mean squared error (MSE) of EB estimators is often used to measure the variability of the estimators. Linearization-based estimators of model MSE as well as jackknife and bootstrap estimators are widely used. On the other hand, National Statistical Agencies are often interested in estimating the design MSE of EB estimators in line with traditional design MSE estimators associated with direct estimators for large areas with adequate sample sizes. Estimators of design MSE of EB estimators can be obtained for area level models but they tend to be unstable when the area sample size is small. Composite MSE estimators are proposed in this paper and they are obtained by taking a weighted sum of the design MSE estimator and the model MSE estimator. Properties of the MSE estimators under the area level model are studied in terms of design bias, relative root mean squared error and coverage rate of confidence intervals. The case of a unit level model is also examined under simple random sampling within each area. Results of a simulation study show that the proposed composite MSE estimators provide a good compromise in estimating the design MSE.