Constrained Bayes in multiplicative area-level models under the precautionary loss function

Consider the problem of benchmarking small-area estimates under multiplicative models with positive parameters. The goal is to propose a loss function that guarantees positive constrained estimates of small-area parameters in this situation. The weighted precautionary loss function is introduced to solve the problem. Compared with the weighted Kullback–Leibler (KL) loss function, our proposed loss function penalizes underestimation of the small-area parameters of interest more for small values of parameters. This property is appealing when we estimate disease rates. It tends to give larger estimates of small-area parameters compared with those obtained under the KL loss function. The hierarchical empirical Bayes and constrained hierarchical empirical Bayes estimates of small-area parameters and their corresponding risk functions under the new proposed loss function are obtained. The performance of the proposed methods is investigated using simulation studies and a real dataset.