In this chapter we shall deal with one of the two basic technical tools required in implementing the EB approach. As mentioned in Chapter 1, EB decision rules can be constructed via two main approaches. The first is based on an explicit estimation of the unknown prior distribution. The second is based on a method of expressing the Bayes estimate or decision rule in terms of functionals of G and estimating the Bayes rule itself directly. We shall also see that, in general, smooth EB rules obtained by the former approach can be ‘better’ than those of the latter. The feasibility of estimating a prior distribution G depends on the possibility of finding a distribution function (d.f.) G satisfying the relationship