A Comprehensive Study of the Bernoulli-Transmuted Geometric Distribution: Theory and Applications

This paper presents the BerTG distribution, an innovative three-parameter discrete probability model developed by convolving a Bernoulli random variable with an independently distributed transmuted geometric random variable. The suggested distribution constitutes a significant and adaptable generalization that en- compasses various established count distributions as specific instances, thereby offering a cohesive framework for modeling a range of count data formats. The BerTG distribution is notable for its exceptional ability to handle many types of dispersion, including as overdispersion, underdispersion, and equidispersion, com- monly found in real-world count data, thereby overcoming a significant weakness of numerous conventional discrete models. A thorough examination of the distributional and structural characteristics of the BerTG model is conducted, including the probability mass function, cumulative distribution function, moments, moment generating function, factorial moments, probability generating function, and index of dispersion, among other aspects. Special emphasis is placed on reliability-theoretic attributes, encompassing the hazard rate function, survival function, reverse hazard rate function, and conditional expectation, which are metic- ulously generated and examined. Additionally, essential actuarial metrics, including the stop-loss premium, value-at-risk, and tail value-at-risk, are analysed to illustrate the model’s appropriateness for risk-theoretic applications. Model parameters are estimated by maximum likelihood estimation, and the asymptotic prop- erties of the resultant estimators are determined. A comprehensive simulation analysis is performed to assess the finite-sample performance of the estimators concerning bias, mean squared error, and consistency across diverse parameter configurations and sample sizes, thereby validating the reliability and accuracy of the estimation technique. The practical applicability of the BerTG distribution is evidenced through real-world data applications, wherein the model is applied to several empirical count datasets displaying diverse dispersion traits. Comparative analyses with various competing discrete distributions demonstrate that the BerTG model consistently attains superior goodness-of-fit performance, as indicated by standard model selection criteria such as the Akaike information criterion, Bayesian information criterion, and chi- square goodness-of-fit statistics. The results combined demonstrate that the BerTG distribution is a very competitive, manageable, and adaptable instrument for the statistical modelling of count data across several applicable fields.