Empirical Bayes estimators for the reproduction parameter of Borel-Tanner distribution

where 0 < θ < 1, r is a positive integer and ar(x) = rx /(x− r)! Initially (1) was derived as the probability distribution of the number of customers served in a queuing system. It also appears in random trees and branching processes. More specifically, it is the distribution of the total progeny in a Galton-Watson process assuming Poisson reproduction, see Aldous [1] for recent applications. Our interest in estimating θ stems from its role as reproduction number of an epidemic infection modeled by a branching process, see Farrington et al. [2]. We study nonparametric (with respect to the prior) empirical Bayes (NPEB) estimators for θ. The NPEB estimation procedures rely on the assumption for existence of a prior distribution G which, however, is unknown. Consider independent copies (X1, θ1), . . . , (Xn+1, θn+1) of (X, θ), where θ has a distribution G, and conditional on θ, X has a Borel–Tanner distribution given by (1). The “past” data consist of independent observations x1, x2, . . . , xn obtained with independent realizations θ1, θ2, . . . , θn of θ, where the Xis are observable and the θis are not observable. Denote by θn(x) an empirical Bayes estimator for θ based on the “past” data and the “present” observation xn+1 = x. As Maritz and Lwin [5] point out, an advantage of using NPEB estimators is the minimum assumptions on the class of prior distributions. It turns out that in the case of Borel-Tanner distribution the Bayes rule assuming LINEX loss depends on the prior through the marginals only. This remarkable