Bivariate count models having one marginal and the other conditionals being of the Poissons form are called pseudo-Poisson distributions. Such models have simple exible dependence structures, possess fast computation algorithms and generate a sufficiently large number of parametric families. It has been strongly argued that the pseudo-Poisson model will be the first choice to consider in modelling bivariate over-dispersed data with positive correlation and having one of the marginal equi-dispersed. Yet, before we start fitting, it is necessary to test whether the given data is compatible with the assumed pseudo-Poisson model. Hence, in the present note we derive and propose a few goodness-of-fit tests for the bivariate pseudo-Poisson distribution. Also we emphasize two tests, a lesser known test based on the supremes of the absolute difference between the estimated probability generating function and its empirical counterpart. A new test has been proposed based on the difference between the estimated bivariate Fisher dispersion index and its empirical indices. However, we also consider the potential of applying the bivariate tests that depend on the generating function (like the Kocherlakota and Kocherlakota and Mu~noz and Gamero tests) and the univariate goodness-of-fit tests (like the Chi-square test) to the pseudo-Poisson data. However, for each of the tests considered we analyse finite, large and asymptotic properties. Nevertheless, we compare the power (bivariate classical Poisson and Com-Max bivariate Poisson as alternatives) of each of the tests suggested and also include examples of application to real-life data. In a nutshell we are developing an R package which includes a test for compatibility of the data with the bivariate pseudo-Poisson model.

翻译：具有一个边际分布和另一个条件分布均为泊松形式的二元计数模型称为伪泊松分布。此类模型具有灵活简单的相依结构、快速的计算算法，并能生成足够多的参数族。已有充分论证表明，在对具有正相关性且其中一个边际分布为等离散的二元过离散数据进行建模时，伪泊松模型将是首选考虑对象。然而，在开始拟合之前，有必要检验给定数据是否与假定的伪泊松模型相容。因此，本文推导并提出了几种针对二元伪泊松分布的拟合优度检验方法。我们重点强调两种检验：一种较为少见、基于估计的概率生成函数与其经验版本之间绝对差值上确界的检验方法；另一种则是基于估计的二元费希尔离散指数与其经验指数之差的检验新方法。同时，我们也考虑了将基于生成函数的二元检验（如Kocherlakota与Kocherlakota检验、Muñoz与Gamero检验）以及一元拟合优度检验（如卡方检验）应用于伪泊松数据的可行性。对于所考虑的每种检验方法，我们均分析了其有限样本、大样本和渐近性质。此外，我们比较了所建议各检验方法的检验功效（以二元经典泊松分布和Com-Max二元泊松分布作为备择假设），并给出了实际数据应用示例。简而言之，我们正在开发一个R语言软件包，其中包含用于检验数据与二元伪泊松模型相容性的检验方法。