Copula-like inference for discrete bivariate distributions with rectangular support

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 the iterative proportional fitting procedure (IPFP), also known as Sinkhorn's algorithm or matrix scaling in the literature, as a sound way to attempt to decompose a bivariate probability mass function into its two univariate margins and a bivariate probability mass function with uniform margins playing the role of a discrete copula. After stating what could be regarded as a discrete analog of Skar's theorem, we investigate, for starting bivariate p.m.f.s with rectangular support, 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 new differentiability result for the iterative proportional fitting procedure which can be of independent interest. Theoretical results are complemented by finite-sample experiments and a data example.