The Asymptotically Optimal Empirical Bayes Estimation about a Class of Uniform Distribution

In this note, we propose a squared error loss empirical Bayes estimator of θ based on past experiences and a present observation X which has conditional distribution. U(θ, cθ+b), where b is an arbitary constant when c>1; b>c when c=1, θ∈Ω=(-b/c-1,∞). When unkown prior G(θ) of θ belongs to the family {G:integral from Ω (θ~2dG(θ)<∞)}, our estimator is asymptotically optimal (see [1]). Let K(x) and k(x) be marginal distribution and density of r. v. X. It is easily seen that