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Pseudo-Poisson Distributions with Nonlinear Conditional Rates

Jared N. Lakhani

TL;DR

This work extends the bivariate pseudo-Poisson framework by incorporating a curved conditional rate $\lambda_2(x_1)=\delta+\beta F(x_1;\boldsymbol{\theta})$ with $F$ drawn from exponential or Lomax families, enabling negative correlations and proper handling of the origin $(0,0)$. It develops exponential- and Lomax-based formulations, derives moments, and provides method-of-moments and maximum-likelihood estimation procedures, along with likelihood-ratio tests to compare sub-models. The authors establish correlation bounds for various sub-models, illustrate parameter-estimation behavior via simulations, and apply the methods to two real datasets, showing improved fit over linear specifications in several cases and highlighting boundary-effects near $(0,0)$. The study concludes that curvature enhances flexibility and interpretability for negatively correlated count data, while noting practical considerations such as estimator existence and model selection via AIC, with opportunities for further goodness-of-fit analysis. Overall, the approach broadens the applicability of bivariate count models by enabling negative dependence and boundary-aware regression-like behavior.

Abstract

Arnold & Manjunath (2021) claim that the bivariate pseudo-Poisson distribution is well suited to bivariate count data with one equidispersed and one overdispersed marginal, owing to its parsimonious structure and straightforward parameter estimation. In the formulation of Leiter & Hamdan (1973), the conditional mean of $X_2$ was specified as a function of $X_1$; Arnold & Manjunath (2021) subsequently augmented this specification by adding an intercept, yielding a linear conditional rate. A direct implication of this construction is that the bivariate pseudo-Poisson distribution can represent only positive correlation between the two variables. This study generalizes the conditional rate to accommodate negatively correlated datasets by introducing curvature. This augmentation provides the additional benefit of allowing the model to behave approximately linear when appropriate, while adequately handling the boundary case $(x_1,x_2)=(0,0)$. According to the Akaike Information Criterion (AIC), the models proposed in this study outperform Arnold & Manjunath (2021)'s linear models.

Pseudo-Poisson Distributions with Nonlinear Conditional Rates

TL;DR

This work extends the bivariate pseudo-Poisson framework by incorporating a curved conditional rate with drawn from exponential or Lomax families, enabling negative correlations and proper handling of the origin . It develops exponential- and Lomax-based formulations, derives moments, and provides method-of-moments and maximum-likelihood estimation procedures, along with likelihood-ratio tests to compare sub-models. The authors establish correlation bounds for various sub-models, illustrate parameter-estimation behavior via simulations, and apply the methods to two real datasets, showing improved fit over linear specifications in several cases and highlighting boundary-effects near . The study concludes that curvature enhances flexibility and interpretability for negatively correlated count data, while noting practical considerations such as estimator existence and model selection via AIC, with opportunities for further goodness-of-fit analysis. Overall, the approach broadens the applicability of bivariate count models by enabling negative dependence and boundary-aware regression-like behavior.

Abstract

Arnold & Manjunath (2021) claim that the bivariate pseudo-Poisson distribution is well suited to bivariate count data with one equidispersed and one overdispersed marginal, owing to its parsimonious structure and straightforward parameter estimation. In the formulation of Leiter & Hamdan (1973), the conditional mean of was specified as a function of ; Arnold & Manjunath (2021) subsequently augmented this specification by adding an intercept, yielding a linear conditional rate. A direct implication of this construction is that the bivariate pseudo-Poisson distribution can represent only positive correlation between the two variables. This study generalizes the conditional rate to accommodate negatively correlated datasets by introducing curvature. This augmentation provides the additional benefit of allowing the model to behave approximately linear when appropriate, while adequately handling the boundary case . According to the Akaike Information Criterion (AIC), the models proposed in this study outperform Arnold & Manjunath (2021)'s linear models.

Paper Structure

This paper contains 58 sections, 6 theorems, 104 equations, 13 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Exponential
  3. Probability mass functions
  4. Moments
  5. Method of moments estimation
  6. Full exponential model:
  7. Case I ($\beta = 1$ or $\beta = -1$):
  8. Case II ($\gamma = 1$):
  9. Case IV ($\gamma = 1$ and $\beta = 1$ or $\gamma = 1$ and $\beta = -1$):
  10. Case V ($\delta = 0$ and $\gamma = 1$):
  11. Correlations
  12. Full exponential model
  13. Case I
  14. $\beta =1$
  15. $\beta = -1$
  16. ...and 43 more sections

Key Result

Theorem 1

Given a pseudo-Poisson form $X_1 \sim \text{Poisson}(\lambda_1)$, $X_2 \mid X_1 \sim \text{Poisson}\left(\lambda_2(x1) \right)$ with $\lambda_1, \lambda_2(x_1) > 0$ for $x_1, x_2 \in \{0, 1, 2\ldots \}$: if $\lambda_2(x_1)$ is strictly increasing with respect to $x_1$, then $\rho(X_1, X_2) > 0$.

Figures (13)

  • Figure 1: Flow diagram of exponential sub-models.
  • Figure 2: $\tilde{\nu}$ vs $M_1$ and $\frac{S_{12}^2}{S_2 - M_2}$ as per Equation \ref{['eq: nu exp']}.
  • Figure 3: $\rho(\alpha,\beta = 1, \gamma = 1, \delta)$ and $\rho(\alpha,\beta = -1, \gamma = 1, \delta)$.
  • Figure 4: Flow diagram of Lomax sub-models.
  • Figure 5: $\tilde{\gamma}'$ vs $M_1$ and $\frac{S_{12}^2}{S_2 - M_2}$ as per Equation \ref{['eq: gamma lomax']}.
  • ...and 8 more figures

Theorems & Definitions (13)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Lemma 1
  • ...and 3 more