Estimation of proportions in small areas: application to the labour force using the Swiss Census Structural Survey

The main objectives of this paper are to find efficient but computationally simple estimators for the proportions of people in the labour force (economic activity rates) in Swiss communes and to estimate their mean‐squared error (MSE) over the sampling replication mechanism (the design MSE). This will be done by combining survey data with administrative data provided by the Swiss Federal Statistical Office. We find estimators with considerably greater efficiency than currently used direct estimators and that are easy to implement. We show that, under a generalized linear mixed model with logit link, the computationally expensive empirical best predictor does not perform appreciably better than a plug‐in estimator. Moreover, for moderate proportions of active workers, the empirical best linear unbiased predictor (EBLUP) based on a much simpler linear mixed model performs similarly to the above estimators. We propose new bootstrap estimators of the design MSE of the EBLUPs, which ‘borrow strength’ similarly to EBLUPs. Realistic simulation studies carried out under both model‐ and design‐based set‐ups indicate great gains in efficiency of the selected small area estimators over the traditional direct estimators and acceptable performance of the proposed bootstrap MSE estimators. In the application using the Swiss data, coefficient‐of‐variation reductions of the estimates obtained for the communes are remarkable.