This study presents a flexible extension of the discrete Gompertz distribution, developed within the framework of the exponentiated geometric model. The resulting formulation, termed the discrete Gompertz exponentiated geometric (DGzExGc) model, enhances the ability of existing discrete distributions to model complex and diverse data structures more effectively. The key mathematical and statistical properties of the model are derived, including the probability mass function, cumulative distribution function, reliability function, and hazard rate function. Additional measures, such as the index of dispersion, skewness, and kurtosis, are explored to assess the model's behavior. Furthermore, entropy and order statistics are examined to provide deeper insights into its structural characteristics. The model accommodates both positively skewed and symmetric distributions, as well as unimodal and bimodal structures, making it highly applicable across various domains. It effectively captures a range of hazard rate functions, such as increasing, decreasing, bathtub-shaped, and increasing-constant trends, which are particularly useful for reliability assessments in medicine, engineering, and environmental studies. A critical feature of sustainable data analysis is the model's ability to handle varying dispersion levels, and the DGzExGc model performs exceptionally well in modeling equi-dispersed, over-dispersed, and under-dispersed count data. Additionally, it demonstrates the ability to represent a range of kurtosis levels, from leptokurtic to platykurtic distributions, proving its robustness in capturing diverse data structures. Parameter estimation is conducted using the maximum likelihood method, and a comprehensive simulation study evaluates the performance of the estimators across different sample sizes. To demonstrate the model's flexibility and applicability, three real-world datasets are analyzed, showcasing its superior adaptability and robustness compared to existing models.
Eliwa et al. (Fri,) studied this question.