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Model-free calibration (Deville and Särndal 1992) is routinely used in official statistics for design-based estimation for sub-populations or domains whose sample sizes are large enough for acceptable precision. As a direct estimation method, model-free calibration becomes unreliable when domain sample sizes get small, leading to unstable estimates. In small area estimation (Rao and Molina 2015), model-based methods such as EBLUP and EB estimators are used. These methods "borrow strength" from related areas via models. Lehtonen and Veijanen (2012) extended the model calibration of Wu and Sitter (2001) to small domain estimation with semi-direct and semi-indirect calibration estimators that are assisted by linear and logistic mixed models. Calibrated weights in model-assisted calibration tend to be more stable than the model-free calibration counterparts, leading to more stable domain estimates. However, the built-in property of model-free calibration to reproduce the published official statistics of auxiliary variables (coherence or benchmarking property) is lost in model-assisted calibration. This property is often appreciated in official statistics. Montanari and Ranalli (2009) proposed multiple model calibration for attaining the coherence property. Lehtonen and Veijanen (2015) introduced a hybrid calibration method for small domain estimation that combines properties of model-free calibration (coherence for a desired set of auxiliary variables) and model calibration (flexible modelling and efficiency improvement). As a further extension, Lehtonen and Veijanen (2017, 2018) proposed two-level hybrid calibration to protect against the possible instability problems in the model-free calibration part of hybrid calibration. In this method, model-assisted calibration operates at the original domain level and the model-free calibration part works at a higher hierarchical level, such as regions that contain the sub-regional areas or domains. Two-level hybrid calibration is proposed as a compromise method, combining model-free calibration and model calibration for design-based small domain estimation. We compare the statistical properties (design bias and accuracy) of the calibration methods by design-based simulation experiments for artificially generated data and real data obtained from statistical registers of Statistics Finland and models of the family of generalized linear mixed models.

References

Deville, J.-C. and Särndal, C.-E. (1992) Calibration estimators in survey sampling. JASA, 87, 376–382.

Lehtonen, R. and Veijanen, A. (2012) Small area poverty estimation by model calibration. Journal of the Indian Society of Agricultural Statistics, 66, 125-133.

Lehtonen, R. and Veijanen, A. (2016) Design-based methods to small area estimation and calibration approach. In: Pratesi, M. (Ed.) Analysis of Poverty Data by Small Area Estimation. Wiley, 109–127.

Lehtonen, R. and Veijanen, A. (2017) A two-level hybrid calibration technique for small area estimation. SAE2017 Conference, Paris, 2017.

Lehtonen, R. and Veijanen, A. (2018) Small area estimation with calibration methods. 10ème Colloque francophone sur les sondages Conference, Lyon, 2018.

Montanari, G. E. and Ranalli, M. G. (2009) Multiple and ridge model calibration. Proceedings of Workshop on Calibration and Estimation in Surveys 2009. Statistics Canada.

Rao, J.N.K. and Molina, I. (2015) Small Area Estimation, 2nd ed. Wiley, New York.

Wu, C. and Sitter, R. R. (2001) A model-calibration approach to using complete auxiliary information from survey data. JASA, 96, 185–193.