The optimal confidence region for a random parameter

Suppose that, under a two-level hierarchical model, the distribution of the vector of random parameters is known or can be estimated well. The data are generated via a fixed, but unobservable, realisation of the vector. We derive the smallest confidence region for a specific component of this random vector under a joint Bayesian/frequentist paradigm. On average this optimal region can be much smaller than the corresponding Bayesian highest posterior density region. The new estimation procedure is especially appealing when one deals with data generated under a highly parallel structure. The new proposal is illustrated with a dataset from a multi-centre clinical study and also with one from a typical microarray experiment. The performance of our procedure is examined via simulation studies. Copyright 2005, Oxford University Press.