Flexible Bivariate Generalized Shifted Inverse Trinomial Distributions for Over- and Under-Dispersed Count Data

Modeling bivariate count data with complex dispersion and dependence structures remains a significant challenge in statistical data analysis. This article introduces two new bivariate count distributions derived from the generalized shifted inverse trinomial distribution. The proposed models, denoted by BGIT-I and BGIT-II, are constructed using convolution and trivariate reduction methods. They provide flexible joint frameworks for modeling correlated count data while accommodating different marginal dispersion patterns. BGIT-I allows negative, near-zero, and positive dependence, whereas BGIT-II induces non-negative dependence through a common component. The proposed models have simple, tractable probability generating functions, which facilitate the derivation of probabilistic properties and motivate a probability-generating-function-based estimation approach alongside maximum-likelihood estimation. The finite-sample performance of the estimators is further examined through a Monte Carlo simulation study. The practical utility of the proposed models is illustrated using two real bivariate count data sets involving shunter accidents and patient counts in critical care and emergency room settings. The results show that the proposed BGIT models provide competitive alternatives for modeling bivariate count data with different dispersion and dependence characteristics.