The discrete Gompertz–Makeham distribution for multidisciplinary data analysis

The Gompertz–Makeham (GM) distribution has the flexibility to model real-world lifetime data with increasing, decreasing, or constant hazard rates, making it exceptionally valuable for applications in survival analysis, actuarial science, demography, and reliability engineering. This study proposes and rigorously analyzes a novel discrete formulation of the classical GM distribution, tailored to address real-world applications where event times are inherently discrete. Utilizing the survival function discretization technique, the authors derive the discrete GM (DGM) model and establish its foundational probability mass function, hazard rate function, and cumulative distribution function. A comprehensive suite of statistical properties—including quantiles, moments, skewness, kurtosis, and order statistics—is developed and examined numerically. Recognizing the challenges of parameter estimation under Type–Ⅱ data censoring, the paper implements both maximum likelihood estimation and Bayesian inference, with the latter incorporating gamma priors and executed via a Metropolis–Hastings Markov chain Monte Carlo algorithm. The paper further evaluates the estimator's performance through extensive simulations. The findings consistently demonstrate the superiority of Bayesian methods, particularly with high posterior density intervals. From three life sciences, several empirical case studies underscore the practical utility of the DGM model, showcasing improved goodness-of-fit relative to existing discrete models, for example, the discrete Nadarajah–Haghighi, discrete modified Weibull, discrete Weibull, and discrete gamma models, among others. Finally, this work fills a notable gap in the literature by extending the GM framework to discrete domains with full inferential machinery.