Empirical bayes estimation of reliability in a positive exponential family

We study empirical Bayes estimation of reliability in a lifetime distribution having probability density f(y|theta)=alpha y(alpha upsilon-1) exp(-y(alpha)/theta)/[Gamma(v)theta(v)], where 0 <theta(1)<theta <theta(2)<infinity for some known constants theta(1) and theta(2). An empirical Bayes estimator (phi(n)) over tilde is constructed and its associated asymptotic optimality is studied. It is shown that (phi(n)) over tilde is asymptotically optimal, and the regret of (phi(n)) over tilde converges to zero at a rates O(ln(2)n/n) (for 0 < v < 1/2 or v =1 cases) or O(ln(2v+v) n/n) (for 1/2 <= v <= 1 or v > 1 and v*=max(1,v) cases), where n is the number of past data available when the estimation problem is considered.