Poisson–QLindley Regression Model: Theory and Applications

Count data are frequently characterized by overdispersion and excess zeros. To address these challenges, we propose a new one-parameter count distribution, called the Poisson–QLindley distribution, obtained through a mixed-Poisson construction using the recently introduced QLindley distribution as the mixing distribution. Unlike existing mixed-Poisson-type extensions that increase model complexity by introducing additional parameters, the proposed model retains a one-parameter structure while providing a greater flexibility for modeling overdispersed count data. Several distributional properties of the proposed model are derived, and parameter estimation is performed by the maximum likelihood method. The distribution is extended to a regression framework through a log-link function that enables the analysis of count responses with explanatory variables. The finite-sample performance of the maximum likelihood estimators is evaluated by Monte Carlo simulation. The proposed models are illustrated using two real datasets and compared with alternative regression models developed for the overdispersion. The results demonstrate that the proposed model provides improved goodness of fit for overdispersed count data with computational simplicity. An interactive R Shiny application is also developed to facilitate estimation, model comparison, and residual diagnostics.