Novel Stein-type Characterizations of Bivariate Count Distributions with Applications

1. Problem Statement

While Stein's method and Stein identities are well-established tools for characterizing univariate distributions (both continuous and discrete) and have been applied to univariate count data (Poisson, Binomial, Negative-Binomial), there is a significant gap in the literature regarding bivariate count distributions.

Existing literature lacks rigorous Stein-type characterizations for the three most common bivariate count models:

1. Bivariate Poisson (BPoi)
2. Bivariate Binomial (BVB)
3. Bivariate Negative-Binomial (BNB)

Without these characterizations, it is difficult to derive sophisticated moment expressions, construct robust goodness-of-fit (GoF) tests, or develop specific tests for symmetry in bivariate count data. The authors aim to fill this gap by deriving these identities and demonstrating their practical utility.

2. Methodology

The authors employ Stein's method, which involves finding a characterizing operator (an identity involving expectations) that uniquely defines a probability distribution.

• Derivation Strategy:

  • For each distribution, the authors start with the known probability generating function (pgf) or the recurrence relations of the probability mass function (pmf).
  • They manipulate these relations to derive linear equations involving expectations of the form $E[X_i f(X_1, X_2)]$.
  • They prove necessity (if the distribution holds, the identity holds) and sufficiency (if the identity holds for all bounded functions $f$, the distribution is uniquely determined).
  • The proofs often involve differentiating the joint pgf with respect to variables and equating coefficients to establish partial differential equations (PDEs) that uniquely solve to the target pgf.

• Application Framework:

  • Moment Calculation: Using the derived identities to create recursive schemes for calculating joint factorial moments and probabilities, avoiding complex closed-form summations.
  • Goodness-of-Fit (GoF) Testing: Constructing test statistics based on the deviation of sample moments from the theoretical Stein identities.
  • Symmetry Testing: Adapting the identities to test if a bivariate distribution is symmetric (i.e., $(X_1, X_2) \stackrel{d}{=} (X_2, X_1)$) by utilizing alternating functions $f(x,y) = -f(y,x)$.

3. Key Contributions

A. Theoretical Derivations

The paper establishes the first Stein characterizations for the three major bivariate count families:

1. Bivariate Poisson (BPoi):

   • Derived identities linking $E[X_1 f(X_1, X_2)]$ and $E[X_2 f(X_1, X_2)]$ to expectations of shifted arguments involving parameters $\lambda_0, \lambda_1, \lambda_2$.
   • Proven to reduce to the classical univariate Stein-Chen identity when one variable is zero.

2. Bivariate Binomial (BVB - Type I):

   • Derived identities involving parameters $N$ (trials) and joint probabilities $p_{ij}$ of the underlying Bernoulli vector.
   • The identities relate expectations of $X_i f(\cdot)$ to expectations of $(N-X_i)f(\cdot)$ with shifted arguments.

3. Bivariate Negative-Binomial (BNB):

   • Derived identities involving the dispersion parameter $\nu$ and probabilities $\pi_0, \pi_1, \pi_2$.
   • The identities link $E[X_i f(\cdot)]$ to terms involving $(\nu + X_i)f(\cdot)$ with shifted arguments.

B. Practical Applications

The authors demonstrate three specific applications of these new identities:

1. Recursive Moment Calculation:

   • Developed recursive schemes for calculating joint probabilities and factorial moments for BVB and BNB distributions.
   • These schemes are computationally superior to existing closed-form formulas, which are often "sophisticated and computationally demanding."

2. Goodness-of-Fit (GoF) Tests:

   • Proposed a new class of test statistics ($T_{1;f}$) based on the Stein identities for BPoi.
   • The statistic measures the deviation of the empirical expectation from the theoretical identity.
   • Simulation Results: The Stein-based tests showed superior power, particularly for detecting underdispersion (BVB alternatives) compared to classical dispersion index tests (like Crockett's index). They maintained correct size (Type I error) at the 5% level.

3. Symmetry Tests:

   • Proposed statistics ($T_{2;f}$ and $T_{3;f}$) to test for symmetry within the BPoi family and general bivariate symmetry.
   • These tests utilize alternating functions (e.g., $f(x,y) = x^a - y^a$).
   • Simulation Results: The $T_{3;f}$ statistic was found to be highly effective for detecting asymmetry within the BPoi family. However, for alternatives outside the BPoi family that are still symmetric, the test correctly failed to reject, highlighting its specificity.

C. Real-World Data Analysis

The authors re-analyzed four real-world datasets (plant counts, injury counts, accident data, and synthetic data) previously studied in the literature.

• Findings: The new Stein-based tests confirmed previous rejections of the BPoi hypothesis for most datasets.
• Insight: The tests provided additional evidence regarding asymmetry. For example, in one dataset, the symmetry test ($T_{3;f}$) detected asymmetry where other tests were less sensitive, suggesting the data originated from an asymmetric BPoi distribution.

4. Results

• Theoretical: Successfully proved that the derived identities uniquely characterize the BPoi, BVB, and BNB distributions.
• Computational: The recursive schemes derived from the identities offer a practical alternative for computing probabilities and moments where closed forms are intractable.
• Statistical Performance:
  • GoF Tests: The Stein-index tests ($T_{1;f}$) outperformed classical dispersion tests, especially in detecting underdispersion (BVB alternatives).
  • Symmetry Tests: The tests effectively distinguished between symmetric and asymmetric distributions. The choice of the weight function $f$ (specifically $f(x,y) = x^{1/2} - y^{1/2}$ vs. $x-y$) influenced power, with $f_{1/2}$ showing better performance for symmetry tests, contrary to GoF tests where $f_1$ was often preferred.

5. Significance

This paper makes a substantial contribution to the field of statistical theory and applied statistics by:

1. Bridging a Theoretical Gap: It extends Stein's method from univariate to bivariate count data, a necessary step for analyzing dependent count variables common in epidemiology, ecology, and reliability engineering.
2. Enabling New Methodologies: It provides the theoretical foundation for developing new, flexible goodness-of-fit and symmetry tests that do not rely on asymptotic approximations of complex likelihood ratios but rather on moment-based identities.
3. Practical Utility: The derived recursive schemes solve a computational bottleneck for calculating moments in complex bivariate models.
4. Future Research Direction: The authors outline a path for future work, including optimizing weight functions for tests, developing non-parametric implementations of symmetry tests, and applying these characterizations to count time-series analysis (e.g., INARMA models).

In summary, Wang and Weiß provide a robust mathematical framework that transforms the theoretical properties of bivariate count distributions into actionable tools for statistical inference and data analysis.