Empirical Bayes estimation for Borel–Tanner distributions

This work deals with empirical Bayes estimation for the reproduction parameter of Borel–Tanner distributions. Losses of two types are considered: squared and linex error losses. For each type of error loss, a nonparametric empirical Bayes estimator is proposed and its associated asymptotic optimality is investigated. It is shown that under a certain condition, the regrets of the proposed empirical Bayes estimators converge to zero at a rate O(n−λ/2)O(n−λ/2) for some number λ,0<λ<2λ,0<λ<2. An example fulfilling the required condition for convergence is also provided.