Measuring uncertainty associated with model-based small area estimators
Section 1. Introduction
Sample survey data are often used to produce estimates of domain (subpopulation) totals or means. Traditional direct estimators for domains, including calibration estimators that use known population totals of auxiliary variables, are designed to provide reliable estimators for domains with large domain-specific sample sizes. However, direct estimators do not provide adequate precision for domains with small sample sizes (called small areas). Yet the demand for reliable small area statistics has greatly increased in recent years. It is therefore necessary to resort to indirect estimators that borrow information from related areas through known auxiliary information such as censuses and administrative records, to increase the efficiency. Indirect estimators based on explicit linking models are widely used; in particular, empirical best (EB) estimators based on area level or unit level linear regression models with random area effects. A detailed account of EB estimation under those models is given by Rao and Molina (2015), Chapters 6 and 7. Section 2 presents EB estimators of small area means under basic area level and unit level models.
EB-type model based estimators are often deemed suitable by National Statistical Agencies to produce official statistics, after careful external evaluations. For example, Beaumont and Bocci (2016) compared EB and direct estimates of unemployment rate for small areas obtained from the Canadian Labour Force Survey (LFS) to “gold standard” estimates obtained from the much larger National Household Survey (comparable to long form census) and found that the relative error of EB estimates is much smaller than the corresponding direct estimates. The authors used a basic area level linear regression model with random area effects to produce EB estimates. External evaluations were first used in the pioneering paper by Fay and Herriot (1979) under a basic area level model to produce estimates of mean income for small places in the United States.
Model mean squared error (MSE) of the EB estimators is often used to measure the variability of the estimators. In particular, linearization-based estimators of model MSE as well as jackknife and bootstrap estimators are widely used. Section 3 gives a brief account of model-based MSE estimation, including estimators based on unconditional and conditional frameworks.
The literature on estimating model MSE is very impressive, but National Statistical agencies are often interested in estimating the design MSE of EB estimators in line with the traditional design MSE estimators of direct estimators for large areas with adequate sample sizes (Pfeffermann and Gilboa, 2017). Estimators of design MSE of EB estimators for the basic area level model can be obtained but they tend to be unstable when the area sample size is small. To address this problem, Section 4 proposes composite MSE estimators obtained by taking a weighted sum of the design MSE estimator and the model MSE estimator. The case of unit level models is also studied under simple random sampling within areas. Section 5 reports the results of simulation studies on the performance of the proposed composite MSE estimators in terms of design absolute relative bias (ARB), relative root mean squared error (RRMSE) and coverage of confidence intervals. Both area level and unit level models are considered in the simulation study. Finally, some conclusions are given in Section 6.
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