Bivariate Pseudo-Poisson Model

• The topic is defined as a joint distribution where one variable is exactly Poisson and the second is modeled by a Poisson regression, enabling both equi-dispersion and over-dispersion.
• It employs linear and nonlinear conditional rate models with closed-form moments and maximum likelihood estimation to capture dependency in count data.
• Extensions include symmetric Poisson conditional models and robust goodness-of-fit tests, with applications in epidemiology, traffic safety, and health statistics.

A bivariate pseudo-Poisson distribution is a class of joint distributions for two non-negative integer-valued random variables $(X_1, X_2)$ in which one marginal distribution is exactly Poisson, and the conditional distribution of the second variable given the first is a Poisson regression (with rates that may be affine or nonlinear in the first variable). This construction enables the modeling of bivariate count data with the property that one margin is equi-dispersed while the other is over-dispersed, and it establishes an explicit, tractable dependence structure driven by the conditional mean specification (Arnold et al., 2020, Lakhani, 18 Nov 2025, Veeranna et al., 2023).

1. Model Specification and Joint Distributions

Linear Pseudo-Poisson Model

Let $X_1, X_2$ be non-negative integer-valued random variables. The canonical three-parameter linear pseudo-Poisson model is defined by:

• $X_1 \sim \mathrm{Poisson}(\lambda_1)$, $\lambda_1 > 0$
• $$X_2\mid X_1 = x_1 \sim \mathrm{Poisson}(\lambda_2 + \lambda_3 x_1)$$, $\lambda_2 \ge 0$, $\lambda_3 \ge 0$

The joint probability mass function (pmf) is

$$P(X_1 = x_1, X_2 = x_2) = e^{-\lambda_1} \frac{\lambda_1^{x_1}}{x_1!} \, e^{-(\lambda_2+\lambda_3 x_1)} \frac{(\lambda_2+\lambda_3 x_1)^{x_2}}{x_2!} \quad\text{for } x_1, x_2 = 0, 1, 2, \ldots$$

The natural parameter space is $$\{\lambda_1 > 0,\, \lambda_2 \ge 0,\, \lambda_3 \ge 0\}$$. Independence is recovered when $\lambda_3 = 0$ (Arnold et al., 2020, Veeranna et al., 2023).

Nonlinear Conditional Rate Extensions

The model can be extended by specifying the Poisson conditional rate via a nonlinear, bounded regression function $F(x_1; \theta)$, yielding (Lakhani, 18 Nov 2025): $$X_2 \mid X_1 = x_1 \sim \mathrm{Poisson}\left(\delta+\beta F(x_1;\theta)\right),\quad \delta\ge 0,\, \beta\neq 0$$ where typically $F(x_1; \theta)$ is non-decreasing, with $F(0;\theta)=0$ and $\lim_{x_1\to\infty} F(x_1; \theta) = 1$. Examples include the exponential kernel $F(x_1;\gamma) = 1-e^{-\gamma x_1}$ and the Lomax kernel $$F(x_1; \gamma', \eta)=1-\left(\tfrac{\gamma'}{x_1+\gamma'}\right)^{\!\eta}$$.

2. Marginal, Conditional, and Generating Functions

• Marginal of $X_1$: By construction, $X_1 \sim \mathrm{Poisson}(\lambda_1)$ (Arnold et al., 2020, Lakhani, 18 Nov 2025).
• Conditional of $X_2 \mid X_1$: $$X_2\mid X_1 = x_1 \sim \mathrm{Poisson}(\lambda_2+\lambda_3 x_1)$$ (linear), or more generally $\mathrm{Poisson}(\lambda_2(x_1))$ (nonlinear) (Arnold et al., 2020, Lakhani, 18 Nov 2025).
• Marginal of $X_2$: $X_2$ is a Poisson mixture, yielding a Neyman Type A marginal with probability generating function (pgf)

$$G_{X_2}(t) = \exp\{\lambda_2 (t-1)\} \cdot \exp\{\lambda_1 (e^{\lambda_3(t-1)}-1)\}$$

for the linear model. Hence $\mathrm{Var}(X_2) > \mathrm{E}(X_2)$ when $\lambda_3 > 0$ (Arnold et al., 2020).

• Joint pgf:

$$G(t_1, t_2) = \exp\Big\{\lambda_1(t_1 e^{\lambda_3(t_2-1)} - 1) + \lambda_2(t_2 - 1)\Big\}$$

Mixed moments can be obtained via differentiation (Arnold et al., 2020, Veeranna et al., 2023).

3. Moments, Dispersion, and Dependence

For the linear model, the first and second moments and covariance are available in closed form: $$\mathrm{E}[X_1]=\lambda_1,\quad \mathrm{Var}[X_1]=\lambda_1$$

$\mathrm{E}[X_2]=\lambda_2+\lambda_3\lambda_1$

$$\mathrm{Var}[X_2]=\lambda_2+\lambda_3\lambda_1+\lambda_3^2\lambda_1$$

$\mathrm{Cov}(X_1, X_2) = \lambda_3 \lambda_1$

$$\rho = \frac{\lambda_3\lambda_1}{\sqrt{\lambda_1(\lambda_2 + \lambda_3\lambda_1+\lambda_3^2\lambda_1)}}$$

Thus, $X_2$ is equi-dispersed for $\lambda_3=0$ and over-dispersed otherwise.

For nonlinear conditional rate models, the sign of the correlation is determined by the sign of $\beta$. The exponential kernel, for instance, yields

$$\mathrm{Cov}(X_1, X_2) = \alpha\beta(1 - \nu)\mu^{\nu-1}$$

with $\mu = e^{\alpha}$, $\nu = e^{-\gamma}$ (Lakhani, 18 Nov 2025).

The generalized dispersion index (Kokonendji–Puig) for $Z=(X,Y)^\top$ can be written

$$\mathrm{GDI}(Z) = \frac{\sqrt{\mathrm{E}[Z]}^\top\,\mathrm{Cov}(Z)\,\sqrt{\mathrm{E}[Z]}}{\mathrm{E}[Z]^\top\mathrm{E}[Z]}$$

with $GDI(Z) > 1$ indicating overdispersion (Veeranna et al., 2023).

4. Parameter Estimation and Inference

Parameter estimation procedures for the pseudo-Poisson model are direct:

• For the linear form, the log-likelihood for i.i.d. observations $\{(x_{1i}, x_{2i})\}$ is

$$\ell(\lambda_1, \lambda_2, \lambda_3) = \sum_i \big[ -\lambda_1 + x_{1i}\ln \lambda_1 - (\lambda_2+\lambda_3 x_{1i}) + x_{2i}\ln(\lambda_2+\lambda_3 x_{1i}) \big]$$

The estimator for $\lambda_1$ is explicit, $\hat{\lambda}_1=\bar{X}_1$, while $\lambda_2$ and $\lambda_3$ are obtained by solving the score equations numerically (e.g., by Newton–Raphson) (Arnold et al., 2020, Veeranna et al., 2023).

• The Fisher information matrix is available in closed form as a function of the parameters and moments of $X_1$.
• Maximum likelihood estimators (MLEs) are consistent and asymptotically normal. Likelihood-ratio tests can be formed for nested submodels (e.g., $\lambda_3=0$ for independence) (Arnold et al., 2020).

For nonlinear pseudo-Poisson models, estimation is also via MLE, with parameter-specific score equations. The estimator for $\alpha$ remains $\hat{\alpha} = n^{-1}\sum_i x_{1i}$ (Lakhani, 18 Nov 2025).

5. Goodness-of-Fit Testing

A range of goodness-of-fit (GoF) tests are available for the pseudo-Poisson model (Veeranna et al., 2023):

Test Class	Principle	Comments
Supremum (pgf-based)	Maximum standardized absolute deviation of empirical vs theoretical pgf over $(t_1,t_2)$ grid	Bootstrap for critical values
Fisher-index–based	Difference between empirical and model-based generalized dispersion index (GDI)	Not consistent vs all alternatives, simple
Muñoz–Gamero quadratic (MG)	Integrated squared deviation of empirical minus model pgf against weight $w(t_1,t_2)$	Bootstrap for critical values
Pointwise KK	Standardized difference at fixed $(t_1, t_2)$	Highly sensitive to location
Classical $\chi^2$	Cell-based comparison of observed and expected counts	Simple, least powerful

Supremum and MG tests are consistent against broad alternatives and show good power properties given moderate sample size ($n \geq 30$). Fisher-index–based tests target overdispersion specifically.

The R package PseudoPoissonGoF (under development) automates fitting and all proposed GoF procedures (Veeranna et al., 2023).

6. Extensions and Generalizations

Symmetric Poisson Conditional Models

Allowing both conditionals to be Poisson (so neither marginal is Poisson except in independence) leads to the symmetric or "bivariate Poisson-conditional" model: $$X \mid Y = y \sim \mathrm{Poisson}(\lambda_1 \lambda_3^y),\quad Y \mid X = x \sim \mathrm{Poisson}(\lambda_2 \lambda_3^x)$$ with $0 < \lambda_3 \leq 1$. The corresponding joint pmf is

$$P\{X = x, Y = y\} = K(\lambda_1, \lambda_2, \lambda_3) \frac{\lambda_1^x \lambda_2^y \lambda_3^{xy}}{x!y!}$$

where $K(\cdot)$ is a normalizing constant requiring summation to infinite limits. Correlation is negative for $\lambda_3 < 1$, and zero under independence ($\lambda_3 = 1$) (Arnold et al., 2023).

Nonlinear Curvature Models

Generalizing the conditional mean to bounded, nonlinear kernels (e.g., exponential, Lomax) enables modeling negative as well as positive correlation, and accommodates complex boundary behaviors as at $(0,0)$. Akaike Information Criterion (AIC) comparisons demonstrate that such models can substantially outperform linear models, especially in cases where negative correlation is present (Lakhani, 18 Nov 2025). For example, setting $\beta<0$ in the exponential kernel induces strictly decreasing conditional means and negative dependence—a feature unavailable in linear forms.

7. Applications and Empirical Results

Pseudo-Poisson models are particularly advantageous where one variable is equi-dispersed (truly Poisson) and the other is over-dispersed. Empirical case studies include:

• Health and Retirement Study: $X_1=$ number of chronic conditions, $X_2=$ health-care utilizations; $X_1$ is nearly equi-dispersed, $X_2$ slightly over-dispersed, estimated parameters $\hat{\lambda}_1=2.643$, $\hat{\lambda}_2=0.688$, $\hat{\lambda}_3=0.031$ (Arnold et al., 2020).
• Interstate-95 Traffic Accidents: $X_1=$ fatalities, $X_2=$ number of injury accidents; both margins over-dispersed, and AIC comparison favors the model that treats $X_2$ as the marginal and $X_1\,|\,X_2$ as Poisson (Arnold et al., 2020).
• Negative Correlation: Simulation studies with exponential-curvature conditional mean (e.g., $\alpha=5, \beta=-20, \gamma=0.5, \delta=25$) fit negative correlations ($\rho\approx-0.6$) that linear pseudo-Poisson models cannot (Lakhani, 18 Nov 2025).

Pseudo-Poisson models are widely used in epidemiology, traffic safety, and health statistics due to analytic tractability, flexibility in dependence structure, and direct parameter interpretation.

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References:

• (Arnold et al., 2020) Statistical Inference for distributions with one Poisson conditional
• (Lakhani, 18 Nov 2025) Pseudo-Poisson Distributions with Nonlinear Conditional Rates
• (Arnold et al., 2023) On classical and Bayesian inference for bivariate Poisson conditionals distributions
• (Veeranna et al., 2023) Goodness of fit tests for the pseudo-Poisson distribution