Bayesian benchmarking of the Fay-Herriot model using random deletion

Benchmarking lower level estimates to upper level estimates is an important activity at the United States Department of Agriculture's National Agricultural Statistical Service (NASS) (e.g., benchmarking county estimates to state estimates for corn acreage). Assuming that a county is a small area, we use the original Fay-Herriot model to obtain a general Bayesian method to benchmark county estimates to the state estimate (the target). Here the target is assumed known, and the county estimates are obtained subject to the constraint that these estimates must sum to the target. This is an external benchmarking; it is important for official statistics, not just NASS, and it occurs more generally in small area estimation. One can benchmark these estimates by "deleting" one of the counties (typically the last one) to incorporate the benchmarking constraint into the model. However, it is also true that the estimates may change depending on which county is deleted when the constraint is included in the model. Our current contribution is to give each small area a chance to be deleted, and we call this procedure the random deletion benchmarking method. We show empirically that there are differences in the estimates as to which county is deleted and that there are differences of these estimates from those obtained from random deletion as well. Although these differences may be considered small, it is most sensible to use random deletion because it does not give preferential treatment to any county and it can provide small improvement in precision over deleting the last one benchmarking as well.