On Improving the Posterior Predictive Distribution of the Difference Between two Independent Poisson Distribution

This paper addresses the exact Bayesian analysis of the difference between two independent Poisson distributions with means μ1 and μ2 respectively, known as the Skellam distribution with parameters (μ1, μ2). We develop a closed form for the posterior predictive distribution of the future distribution under the order constraint of μ1 > μ2. This kind of constraint is quite common and useful in applications specially in sports data analysis. We show that the proposed distribution estimator outperforms other types of distribution estimators in the literature. We use a simulation study with an example regarding the prediction in soccer games to show the performance of the proposed method.