BIVARIATE EXTENSIONS OF SKELLAM'S DISTRIBUTION

Abstract

Skellam's name is traditionally attached to the distribution of the difference of two independent Poisson random variables. Many bivariate extensions of this distribution are possible, e.g., through copulas. In this paper, the authors focus on a probabilistic construction in which two Skellam random variables are affected by a common shock. Two different bivariate extensions of the Skellam distribution stem from this construction, depending on whether the shock follows a Poisson or a Skellam distribution. The models are nested, easy to interpret, and yield positive quadrant-dependent distributions, which share the convolution closure property of the univariate Skellam distribution. The models can also be adapted readily to account for negative dependence. Closed form expressions for Pearson's correlation between the components make it simple to estimate the para-meters via the method of moments. More complex formulas for Kendall's tau and Spearman's rho are also provided.