A Generalized Nabla Geometric Distribution and Its Practical Utility in Engineering Data

In this study, we introduce a novel three-parameter discrete probability distribution, defined by the parameters α, β, and λ, called the Generalized Nabla Geometric Distribution (GNGD). The proposed model is constructed through the discretization of a generalized mixture of exponential distributions. Among its parameters, β plays a significant role in governing the tail behavior and determining the shape of the hazard rate function. The adaptability of the GNGD is further highlighted by its capability to model under-dispersed, equi-dispersed, and over-dispersed data. Several important theoretical properties of the proposed distribution are explored, including the probability generating function, moments, and reliability characteristics, all obtained in explicit closed forms. In addition, the distribution is characterized, and the asymptotic behavior of its minimum and maximum order statistics is investigated. Parameter estimation is performed using the maximum likelihood estimation (MLE) approach, and an extensive simulation study is conducted to evaluate the efficiency and consistency of the estimators. To illustrate its practical usefulness, the GNGD is fitted to two real engineering data sets arising from repairable and non-repairable systems. The findings demonstrate that the proposed distribution provides an excellent fit to both data sets, achieving the smallest information loss criteria and the largest p-values among the competing goodness-of-fit measures considered.