Empirical Bayes estimation of theta(b) in positive exponential families

摘要

Consider a positive exponential family having probability density f(y vertical bar theta)=u(y)beta(theta)exp(-y/theta), y>0, theta>0.With suitable values of b and c, the parameter c theta(b) may denote the mean, the variance or the hazard rate of the probability distribution. In this paper, we study the empirical Bayes estimation of the parameter theta(b) for any fixed real value b. Two empirical Bayes estimators (phi) over cap (n) an phi(n)* are constructed according to the prior information about the parameter space Omega=(0,infinity) Omega=(theta(1),theta(2)), where 0(n) is asymptotically optimal, having rates of convergence O((In n)(2(lambda s-2)/lambda s)((lambda s-2)/lambda s)) or O((In-2 n)((1-b)lambda-1/2s)((lambda s-2)/lambda s)), depending on b > 0 or b < 0 where s > 2 and lambda is positive number such that 2/s 0 or b < 0.