Admissibility of MLE for simultaneous estimation in negative binomial problems

We consider the problem of estimating θ = (θ₁,...,θ_p) under the weighted squared error loss when the observations x_i, i = 1, 2, ..., p are independently from negative binomial distributions NB(r_i, θ_i). There has been considerable interest in providing the Stein type estimators for negative binomial parameter θ. Hwang has shown that the UMVUE is inadmissible when p ≥ 3 under certain weighted squared error loss. It is therefore natural to ask about the admissibility of the MLE estimator. In this paper we address this problem and prove that, under the weighted squared error loss, where the weights are positive and bounded, the MLE is admissible for all p through a stepwise Bayes argument.