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In this paper, we treat estimation and prediction problems where negativemultinomial variables are observed and in particular consider unbalancedsettings. First, the problem of estimating multiple negative multinomialparameter vectors under the standardized squared error loss is treated and anew empirical Bayes estimator which dominates the UMVU estimator under suitableconditions is derived. Second, we consider estimation of the joint predictivedensity of several multinomial tables under the Kullback-Leibler divergence andobtain a sufficient condition under which the Bayesian predictive density withrespect to a hierarchical shrinkage prior dominates the Bayesian predictivedensity with respect to the Jeffreys prior. Third, our proposed Bayesianestimator and predictive density give risk improvements in simulations.Finally, the problem of estimating the joint predictive density of negativemultinomial variables is discussed.

Bayesian Shrinkage Approaches to Unbalanced Problems of Estimation and Prediction on the Basis of Negative Multinomial Samples