A Matrix Approach to Longevity Calculations for the Gamma-Gompertz and Related Models

Hal Caswell, Woods Hole Oceanographic Institution

The gamma-Gompertz model assumes fixed proportional frailty applied to a Gompertz baseline mortality schedule, with a gamma-distributed initial frailty. Analytical expressions for the longevity statistics of this model are mathematically challenging (e.g., Missov 2013 for life expectancy). I present a matrix formulation that provides ready computation of statistics including the mean, variance, and skewness (indeed, all moments) of longevity at all ages, the joint distribution of age and frailty at death, the life disparity, and the dynamics of the frailty distribution. The model is a vec-permutation formulation of an absorbing Markov chain, and makes it easy to study the response to the parameters of the Gompertz function and the variance of frailty. This, inter alia, partitions variance of longevity into components due to individual stochasticity and heterogeneous frailty. The model extends easily to any baseline mortality, any type of frailty effect, and to dynamic as well as fixed heterogeneity.