A novel probability distribution for count data modelling based on Bernoulli–Poisson-Ailamujia convolution: properties, regression framework and applications

The convolution of independent random variables serves as a powerful methodological framework for developing flexible probability distributions. This study uses this methodology to introduce the Bernoulli-Poisson-Ailamujia distribution. The new model can accommodate both overdispersed and underdispersed count data simultaneously. We derive various statistical properties of the new distribution, including moments, index of dispersion, mode, skewness, and kurtosis. We also study parameter estimation using maximum likelihood estimation and assess the finite-sample behavior of the estimators using Monte Carlo simulations. Furthermore, we also develop a flexible count regression model based on the Bernoulli-Poisson-Ailamujia distribution to investigate the relationship between covariates and count responses. Applications of the new model to three real-world datasets demonstrate its superiority over competing models.