Novel Stein-type Characterizations of Bivariate Count Distributions with Applications

2026-02-27

AI Summary

This study addresses a critical gap in the Stein method literature by developing novel Stein identities for bivariate Poisson, binomial, and negative binomial distributions—distributions for which such characterizations were previously unavailable. By integrating probabilistic identity derivations with hypothesis testing theory, the work systematically extends the applicability of Stein’s method to multivariate discrete settings. The proposed framework not only yields explicit expressions for higher-order moments but also enables flexible goodness-of-fit and symmetry tests. Empirical validation on real-world data demonstrates the practical utility and effectiveness of the approach, thereby broadening the methodological scope of Stein techniques in statistical inference for discrete multivariate data.

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Abstract

The derivation and application of Stein identities have received considerable research interest in recent years, especially for continuous or discrete-univariate distributions. In this paper, we complement the existing literature by deriving and investigating Stein-type characterizations for the three most common types of bivariate count distributions, namely the bivariate Poisson, binomial, and negative-binomial distribution. Then, we demonstrate the practical relevance of these novel Stein identities by a couple of applications, namely the deduction of sophisticated moment expressions, of flexible goodness-of-fit tests, and of novel tests for the symmetry of bivariate count distributions. The paper concludes with an analysis of real-world data examples.

Problem

• Stein characterization
• bivariate count distributions
• goodness-of-fit tests
• symmetry testing
• moment expressions

Innovation

• Stein characterization
• bivariate count distributions
• goodness-of-fit test
• moment identities
• symmetry test