Nearly minimax empirical Bayesian prediction of independent Poisson observables

We consider simultaneous predictive distributions for independent Poisson observables and evaluate the performance of predictive distributions using the Kullback--Leibler (K-L) loss. We propose a class of empirical Bayesian predictive distributions that dominate the Bayesian predictive distribution based on the Jeffreys prior. The K-L risk of the empirical Bayesian predictive distributions is demonstrated to be less than $1.04$ times the minimax lower bound.