Bayesian Shrinkage Estimation of Negative Multinomial Parameter Vectors

Abstract

The negative multinomial distribution is a multivariate generalization of the negative binomial distribution. In this paper, we consider the problem of estimating an unknown matrix of probabilities on the basis of observations of negative multinomial variables under the standardized squared error loss. First, a general sufficient condition for a shrinkage estimator to dominate the UMVU estimator is derived and an empirical Bayes estimator satisfying the condition is constructed. Next, a hierarchical shrinkage prior is introduced, an associated Bayes estimator is shown to dominate the UMVU estimator under some conditions, and some remarks about posterior computation are presented. Finally, shrinkage estimators and the UMVU estimator are compared by simulation.

31 pages; the code for numerical computation of the hierarchical Bayes estimator in Section 4 has been corrected; Tables 2, 3, and 4 and the second-to-the-last paragraph of Section 4 have been changed