Unified Stein‐Type Characterizations of Bivariate Count Distributions With Applications

The derivation and application of Stein identities have received considerable research interest, especially for continuous distributions and univariate discrete distributions. In this article, we derive a unified Stein‐type characterization for three bivariate count models, namely the bivariate Poisson, type‐I bivariate binomial, and bivariate negative‐binomial distributions. A single affine logarithmic‐derivative equation yields the three distribution‐specific identities. We then illustrate several applications, including probability and moment recursions, goodness‐of‐fit tests, and tests concerning the symmetry of bivariate count distributions. The paper concludes with analyses of real‐world data examples.