A General Approach to Nonparametric Empirical Bayes Estimation

Let (X1,θ1), (X2, θ2), … ,(X N , θ N ), (X N + 1, θ N + 1) be independent random vectors with each θi distributed according to some unknown prior density g. Given θi, let Xi have the conditional density qi(xθi), i = 1, … , N + 1. In each pair the first component is observable, but the second is not. The objective is to estimate a known function b(θ N + 1) of θ N + 1 A general technique for construction of empirical Bayes estimators of b(θ N + 1) is proposed and their convergence rates are examined. The special case, when the conditional densities qi(x/θ), i = 1, … , N + 1, are identical, is also discussed. The theory is used to estimate of some reliability characteristics of nuclear power plant equipment.