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} 提出一个有力论据，主张采用\cite{Csi75}意义上的$I$-投影作为分解二元概率质量函数（p.m.f.）的合理方法：将其分解为两个一元边际分布与一个以均匀边际充当离散Copula角色的二元p.m.f.。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算法（IPFP）实现，该算法在文献中亦称为Sinkhorn算法或矩阵缩放。在给出二元p.m.f.可按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p.m.f.，研究了非参数与参数估计方法以及底层离散Copula的拟合优度检验。相关渐近结果基于Fr\'echet类上$I$-投影的可微性结论给出，该结论本身或具有独立研究价值。理论结果通过有限样本实验与数据实例加以补充。