On the Generalized Gompertz and Reversed Generalized Gompertz Distributions

This paper presents a comprehensive mathematical and statistical analysis of the generalized Gompertz and reversed generalized Gompertz distributions, introduced by [W. J. Willemse and R. Kaas, Rational reconstruction of frailty-based mortality models by a generalization of Gompertz’ law of mortality, Insur. Math. Econ. 40 (2007) 468–484]. A wide range of mathematical properties is rigorously derived, including the shapes of the density functions, moment generating and characteristic functions, ordinary and conditional moments, quantiles, mean deviations, mean residual life, expected inactive time, inequality curves (Lorenz, Bonferroni, Zenga), order statistics, extreme value limits, and Kullback–Leibler divergence measures (Shannon, Rényi, Tsallis, Kullback–Leibler) and the distributions of the sums and differences of generalized Gompertz and reversed generalized Gompertz random variables. The maximum likelihood method is employed for parameter estimation, and the asymptotic properties of the estimators are established. A detailed simulation study demonstrates the finite-sample performance of the estimators, confirming their consistency, low bias, and adequate coverage probabilities across various parameter configurations and sample sizes. Finally, the applicability of both distributions is illustrated using a real data set on lifetimes of glass capacitors, where both models provide satisfactory fits and outperform a range of standard competitors. The results indicate that the distributions are flexible and theoretically well-founded models suitable for applications in reliability.