Gompertz-Makeham Life Expectancies – Analytical Solutions, Approximations, and Inferences Trifon I. Missov 1 , Adam Lenart 1 , and James W. Vaupel 1 1 Max Planck Institute for Demographic Research Abstract We study the Gompertz and Gompertz-Makeham mortality models. We prove that the resulting life expectancy can be expressed in terms of a hypergeometric function if the population is heterogeneous with gamma-distributed individual frailty, or an incom- plete gamma function if the study population is homogeneous. We use the properties of hypergeometric and incomplete gamma functions to construct approximations that allow calculating the respective life expectancy with high accuracy and interpreting the impact of model parameters on life expectancy. Introduction Parametric models of human mortality date back to Gompertz (1825) and his perception of mortality rates that grow exponentially with age. Makeham’s contribution (Makeham 1860) consists in the addition of an age-independant constant that, on the one hand, accounts for mortality that is not related to aging and, on the other hand, introduces an additional third parameter that improves the model fit. In human populations, the observed overestimation of the Gompertz-Makeham model at ages above 80 demanded the study of models that account for unobserved heterogeneity (Beard 1959). Vaupel et al. (1979) introduced a random variable, called frailty, that modulates individual lifetimes. The resulting mixture of two distributions, one for the general mortality schedule and one for frailty, describes the process at population level. The simplest model (Vaupel et al. 1979) that accurately captures the observed mortality dynamics (Missov and Vaupel 2011) is the Gamma-Gompertz (or Gamma-Gompertz-Makeham) model. Within its framework individual frailty Z is described by a p.d.f. π(z )= λ k Γ(k) z k−1 e −λz , k, λ > 0 1