An empirical bayes estimation problem with nonidentical components involving normal distributions

Abstract

Let {a_n}, {b_n}, and be known sequences of real numbers with a_n ≠ 0 and and let {(θ_n, X_n)} be a sequence of independent random vectors where the θ_n are iid G and unobservable and, given θ_n = θ, X_n has the univariate normal density with mean a_n θ + b_n and variance The first part of the paper exhibits estimators for the density of X_n and its kth derivative, k = 1,2,…, and obtains rates at which the mean-squared errors go to tsro Then, with R_n+1(G) denoting the infinum Bayes risk for the squared error loss estimation of θ_n+1 using X_n+1, the second part of the paper exhibits asymptotically optimal empirical Bayes rules t_n (X₁,…, X_n; X_n+1) and obtains rates at which E(t_n − θ_n+1)² −R_n+1(G)−0.

Keywords:

• empirical Bayes
• squared error loss estimation
• density estimation