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Novel Stein-type Characterizations of Bivariate Count Distributions with Applications

Shaochen Wang, Christian H. Weiß

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.

Novel Stein-type Characterizations of Bivariate Count Distributions with Applications

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.
Paper Structure (10 sections, 3 theorems, 75 equations, 3 tables)

This paper contains 10 sections, 3 theorems, 75 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Stein Characterization of Bivariate Poisson Distribution
  3. Stein Characterization of Bivariate Binomial Distribution
  4. Stein Characterization of Bivariate Negative-binomial Distribution
  5. Applications
  6. Moment Calculations
  7. Goodness-of-fit Testing
  8. Testing for Symmetry
  9. Real-World Data Analyses
  10. Conclusions

Key Result

Theorem 1

Suppose $(X_1,X_2)\sim \mathrm{BPoi}(\lambda_0; \lambda_1,\lambda_2)$, then for any function $f: \mathbb{N}_0\times\mathbb{N}_0\to\mathbb{R}$ such that the following expectation exist, the following two identities hold: Conversely, if the identities thm-eq-1--thm-eq-2 hold for all $f: \mathbb{N}_0\times\mathbb{N}_0\to\mathbb{R}$, and assuming that $(X_1,X_2)$ is a non-negative integer-valued rand

Theorems & Definitions (13)

  • Theorem 1: Stein characterization of bivariate Poisson distribution
  • proof
  • Remark 1
  • Theorem 2: Stein characterization of bivariate binomial distribution
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 3: Stein characterization of bivariate NB distribution
  • proof
  • ...and 3 more