On the Bivariate Skellam Distribution

Abstract

In this paper, we introduce a new distribution on Z 2 , which can be viewed as a natural bivariate extension of the Skellam distribution. The main feature of this distribution a possible dependence of the univariate components, both following univariate Skellam distributions. We explore various properties of the distribution and investigate the estimation of the unknown parameters via the method of moments and maximum likelihood. In the experimental section, we illustrate our theory. First, we compare the performance of the estimators by means of a simulation study. In the second part, we present two applications to a real data set and show how an improved fit can be achieved by estimating mixture distributions.

Bivariate Extensions of Skellam's Distribution

“…It is clear that this construction meets condition (2). In other words, the parameter θ does not affect the marginal distributions of X 1 and X 2 .…”

“…The desired expression for cov(X 1 , X 2 2 ) results from the fact that E(Y 0 ) = θ 1 − θ 2 , E(Y 2 ) = λ 21 − λ 22 − (θ 1 − θ 2 ), var(Y 0 ) = θ 1 + θ 2 , and E(Y 3 0 ) = (θ 1 − θ 2 ) + 3(θ 2 1 − θ 2 2 ) + (θ 1 − θ 2 ) 3 . The formula for cov(X 2 1 , X 2 ) can be obtained in a similar way.…”

A second look at inference for bivariate Skellam distributions

“…This model, originally due to Skellam (), has recently attracted renewed interest through various applications, most notably to sports (Karlis & Ntzoufras ; ). These developments motivated Genest and Mesfioui () and Bulla et al () to propose different possible bivariate extensions of the Skellam distribution.…”

Thinning-based models in the analysis of integer-valued time series: a review

“…For this case, the authors assume that Z t is modeled through a bivariate Skellam distribution (Bulla et al, 2014).…”