A flexible count data model based on Bernoulli-Poisson-geometric convolution

The current work introduces a new three-parameter discrete distribution for modeling over-dispersed as well as under-dispersed count data. The proposed distribution is derived as the convolution of the geometric distribution and the recently introduced BerPoi distribution. Here the proposed model is identified as the BPG model due to its genesis. We explore important statistical properties of the BPG distribution, including moments, survival and hazard rate functions, recurrence relations, generating functions, and dispersion behavior. We also establish attractive characterizations of this distribution in terms of conditional expectation and the reversed hazard rate function. The maximum likelihood estimation method is employed to estimate the unknown parameters of the distribution, and extensive simulation experiments are performed to assess its performance. A flexible count data regression model based on the proposed distribution is also developed. Different real-world applications in modeling over- and under-dispersed count data with and without covariates are considered to establish the relevance of the proposed model. The BPG model has convenient statistical properties and is shown to be better than its closest competitors in modeling applications.