The Gompertz force of mortality in terms of the modal age at death
Demographic Research, 32:36, 1031–1048 (2015)
The Gompertz force of mortality (hazard function) is usually expressed in terms of a, the initial level of mortality, and b, the rate at which mortality increases with age.
We express the Gompertz force of mortality in terms of b and the old-age modal age at death M and present similar relationships for other widely-used mortality models. Our objective is to explain the advantages of using the parameterization in terms of M.
Using relationships among life table functions at the modal age at death, we derive the Gompertz force of mortality as a function of the old-age mode. We estimate the correlation between the estimators of old (a and b) and new (M and b) parameters from simulated data.
When the Gompertz parameters are statistically estimated from simulated data, the correlation between estimated values of b and M is much less than the correlation between a and b. For the populations in the Human Mortality Database there is a negative association between a and b and a positive association between M and b.
Using M, the old-age mode, instead of a, the level of mortality at the starting age, has two major advantages. First, statistical estimation is facilitated by the lower correlation between the estimators of model parameters. Second, estimated values of M are more easily comprehended and interpreted than estimated values of a.
Keywords: age at death, correlation, demographic models, mathematical demography, maximum likelihood method, statistical analysis