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