ASYMPTOTICALLY OPTIMAL EMPIRICAL BAYES ESTIMATORS OF CONTINUOUS FUNCTIONS OF THE PARAMETER IN ONE DIMENSIONAL DISCRETE EXPONENTIAL FAMILY
Bei Tao · 1982
Consider a family of discrete exponential distribution in the formP_(?)(X(?)x)(?)h(x)β(θ)θ~(?),x(?)0,1,2,…,θ∈(?),(?)={θ:θ0,sum from x=0 to ∞ h(x)θ~x∞}.Let f(θ)be a continuous function on (?),and{I_k}be a sequence of bounded and closedintervals satisfying the condition (?).Let (?) be a class of prior distributions on (?),whose member G satisfies the conditionsintegral from(?)(θ)dG(θ)∞ and (?)M_k~2P_G(I_k~(?))=0,where I_k~(?)-I_k and M_k(?)|f(θ)|.Under these conditions,we haveTheorem.Let1°h(x)0,x=0,1,2,…,2°There exists a constant A such thath~2(x)≤Ah(x-1)h(x+1),x=1,2,…Then an asymptotically optimal empirical Bayes estimator of f(θ)relative to the class (?)of prior distributions can be constructed.