Enhanced regression modelling for both under- and overdispersed count data

Abstract In this study, we propose a novel regression model for under- and overdispersed count data based on the Complex Triparametric Pearson ( CTP ) distribution. This model overcomes the limitations of existing models, such as the Conway–Maxwell–Poisson or the hyper-Poisson regression models. The CTP model offers the advantage of adjusting the dispersion based on the covariates considered for the mean, thus avoiding overfitting. Additionally, the log-likelihood function of the CTP model can be expressed in terms of the mean, facilitating its optimization and significantly reducing the computational calculation time compared to the aforementioned models where there are no explicit expressions for the probability mass function and moments in terms of the parameters of the underlying distribution, requiring the use of numerical approximations. The development of the CTP model is presented, followed by a simulation study and illustrative examples of its application. We conclude by highlighting the advantages of the proposed model in terms of flexibility, precision, and computational efficiency.