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.

arXiv comment31 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

References in corpus (2)

• Simultaneous prediction of independent Poisson observables
• Simultaneous prediction for independent Poisson processes with different durations

Cited by in corpus (6)

• Moments of the negative multinomial distribution
• Bayesian Estimation for the Multivariate Hypergeometric Distribution Incorporating Information from Aggregated Observations
• Bayesian Estimation of Multinomial Cell Probabilities Incorporating Information from Aggregated Observations
• Bayesian Shrinkage Estimation for Stratified Count Data
• Bayesian Shrinkage Approaches to Unbalanced Problems of Estimation and Prediction on the Basis of Negative Multinomial Samples
• Shrinkage priors for nonparametric Bayesian prediction of nonhomogeneous Poisson processes