After reviewing a large body of literature on the modeling of bivariate discrete distributions with finite support, \cite{Gee20} made a compelling case for the use of $I$-projections in the sense of \cite{Csi75} as a sound way to attempt to decompose a bivariate probability mass function (p.m.f.) into its two univariate margins and a bivariate p.m.f.\ with uniform margins playing the role of a discrete copula. From a practical perspective, the necessary $I$-projections on Fr\'echet classes can be carried out using the iterative proportional fitting procedure (IPFP), also known as Sinkhorn's algorithm or matrix scaling in the literature. After providing conditions under which a bivariate p.m.f.\ can be decomposed in the aforementioned sense, we investigate, for starting bivariate p.m.f.s with rectangular supports, nonparametric and parametric estimation procedures as well as goodness-of-fit tests for the underlying discrete copula. Related asymptotic results are provided and build upon a differentiability result for $I$-projections on Fr\'echet classes which can be of independent interest. Theoretical results are complemented by finite-sample experiments and a data example.

翻译：在综述大量关于有限支撑二元离散分布建模的文献后，\cite{Gee20} 有力地论证了采用 \cite{Csi75} 意义上的 $I$-投影作为合理方法，将二元概率质量函数分解为两个单变量边缘分布和一个充当离散Copula角色的均匀边缘二元概率质量函数。从实践角度看，Fr\'echet类上的必要 $I$-投影可通过迭代比例拟合过程（即文献中的Sinkhorn算法或矩阵缩放）实现。在给出二元概率质量函数可依上述方式分解的条件后，我们针对具有矩形支撑的初始二元概率质量函数，研究了非参数与参数估计方法，以及相应离散Copula的拟合优度检验。相关渐近结果基于Fr\'echet类上 $I$-投影的可微性结论，该结论本身可能具有独立研究价值。理论结果通过有限样本实验与数据实例进行补充。