Empirical Bayes estimators in hierarchical models with mixture priors

We consider subgroup analyses within the framework of hierarchical modeling and empirical Bayes (EB) methodology for general priors, thereby generalizing the normal–normal model. By doing this one obtains greater flexibility in modeling. We focus on mixture priors, that is, on the situation where group effects are exchangeable within clusters of subgroups only. We establish theoretical results on accuracy, precision, shrinkage and selection bias of EB estimators under the general priors. The impact of model misspecification is investigated and the applicability of the methodology is illustrated with datasets from the (medical) literature.