On the Gompertz–Makeham law: A useful mortality model to deal with human mortality

The Gompertz-Makeham model was introduced as an extension of the Gompertz model in the second half of the 19th century by the British actuary William M. Makeham. Since then, this model has been successfully used in biology, actuarial science, and demography to describe mortality patterns in numerous species (including humans), determine policies in insurance, establish actuarial tables and growth models. In this paper, we derive some structural properties of the Gompertz-Makeham model in statistics, demography, and actuarial sciences, and present some other ones already introduced in the literature. All structural properties we provide are expressed in closed-form, which eliminates the need to evaluate them with numerical integration directly. In addition, we study the estimation of the Gompertz-Makeham model parameters through the discrete Poisson and Bell distributions. In particular, we verify that the recently introduced discrete Bell distribution can be an interesting alternative to the Poisson distribution, mainly because it is suitable to deal with overdispersion, unlike the Poisson distribution. On the basis of real mortality datasets, we compute the remaining life expectancy for several countries and verify that the Gompertz-Makeham model, especially under the Bell distribution, provides proper results to deal with human mortality in practice.