Analytical solutions and approximations to Gompertz-Makeham life expectancies

Adam Lenart, Max Planck Institute for Demographic Research
Trifon I. Missov, Max Planck Institute for Demographic Research

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 incomplete 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.