A Generalized Calibration Approach Ensuring Coherent Estimates with Small Area Constraints

Within this article, a generalized calibration approach is presented, which provides coherent and efficient estimates considering a high number of constraints on different hierarchical levels. These constraints may be obtained from different sources such as survey data, register data, administrative data, or even other sources like big data derived using different estimation approaches, including small area techniques on different levels of interest. In order to incorporate a possible heterogeneous quality and the multitude of the constraints, a relaxation of selected constraints is proposed. In that regard, predefined tolerances are assigned to hardly achievable constraints, mostly at low aggregation levels, or sample estimates with non-negligible variances. In addition, the presented generalized calibration approach allows the use of box-constraints for the calibration weights in order to avoid an inappropriate high variation of the resulting weights. Furthermore, various penalty functions are presented in order to accommodate particular circumstances in applications. The proposed iterative algorithm provably finds the optimal solution and the numerical implementation is able to deal with a huge data base such as the set of all households in Germany. The performance is demonstrated in a short simulation study.