Empirical Bayes estimation of θbθb in positive exponential families

Consider a positive exponential family having probability density f(y|θ)=u(y)β(θ)exp(-y/θ)f(y|θ)=u(y)β(θ)exp(-y/θ), y>0y>0, θ>0θ>0. With suitable values of b and c  , the parameter cθbcθb 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 θbθb for any fixed real value b  . Two empirical Bayes estimators ϕ˜n and ϕn* are constructed according to the prior information about the parameter space Ω=(0,∞)Ω=(0,∞) or Ω=(θ1,θ2)Ω=(θ1,θ2), where 0<θ1<θ2<∞0<θ1<θ2<∞ are known constants. The asymptotic optimality of the proposed empirical Bayes estimators is investigated. The rates of convergence of the associated regrets are established. It has been shown that under certain conditions, ϕ˜n is asymptotically optimal, having rates of convergence O((lnn)2(λs-2)/λs/n(λs-2)/λs)O((lnn)2(λs-2)/λs/n(λs-2)/λs) or O((ln2n)(1-b)λ-1/2s/n(λs-2)/2s), depending on b>0b>0 or b<0b<0 where s>2s>2 and λλ is positive number such that 2/s<λ<2(1-1/s)2/s<λ<2(1-1/s); and ϕn* is asymptotically optimal, having rates of convergence O(ln2n/n) or O((lnn)2(1-b)+1/n)O((lnn)2(1-b)+1/n), depending on b>0b>0 or b<0b<0.