A General Approach to Nonparametric Empirical Bayes Estimation

Abstract

Let (X₁,θ₁), (X₂, θ₂), … ,(X _N , θ _N ), (X _{N + 1}, θ _{N + 1}) be independent random vectors with each θ_i distributed according to some unknown prior density g. Given θ_i, let X_i have the conditional density q_i(xθ_i), 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 q_i(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.

Keywords:

• Empirical Bay estimation
• posterior quadratic risk
• convergence rate
• reliability characteristics