Pseudo-Poisson Distributions with Nonlinear Conditional Rates

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.