Small area estimation has become a topic of growing importance in recent years. Model-based small area estimates, when aggregated, need not correspond to the direct survey estimate for a larger area, e.g., national, which may be a cause of concern if (i) the sample size for the larger area is sufficiently large that the direct estimate is regarded as reliable, and/or (ii) the direct estimate for the larger area has any sort of official status. Substantial deviation of an aggregate of model-based small area estimates from the corresponding direct estimate for a large area may also suggest model failure. These considerations motivate benchmarking, which is some form of calibration that adjusts individual area level estimates so they aggregate to a direct estimate for a large area.
In this talk, we consider benchmarking issues in the context of small area estimation. We find optimal estimators within the class of benchmarked linear estimators under either external or internal benchmark constraints. We extend existing results for both external and internal benchmarking, and also provide some links between the two. In addition, we obtain necessary and sufficient conditions for self-benchmarking for an augmented model. Most of our results are found using ideas of orthogonal projection.