Gompertz–Makeham law of mortality

Gompertz–Makeham
Parameters	${\displaystyle \alpha \in \mathbb {R} ^{+}}$ ${\displaystyle \beta \in \mathbb {R} ^{+}}$ ${\displaystyle \lambda \in \mathbb {R} ^{+}}$
Support	${\displaystyle x\in \mathbb {R} ^{+}}$
PDF	${\displaystyle \left(\alpha e^{\beta x}+\lambda \right)\cdot \exp \left[-\lambda x-{\frac {\alpha }{\beta }}\left(e^{\beta x}-1\right)\right]}$
CDF	${\displaystyle 1-\exp \left[-\lambda x-{\frac {\alpha }{\beta }}\left(e^{\beta x}-1\right)\right]}$

The Gompertz–Makeham law of mortality is a mathematical model for the age pattern of death rates. It expresses the force of mortality (instantaneous hazard of death) as the sum of two components: an age-dependent term, the Gompertz function, that increases approximately exponentially with age, and an approximately age-independent background term known as the Makeham term. In populations where deaths from external causes are rare, the background component is often small and mortality can be well described by the simpler Gompertz law of mortality.

For adult humans, the Gompertz component captures the empirical regularity that the individual risk of death rises steeply with age. After early adulthood, many populations are reasonably described by an almost exponential increase in hazard, with the probability of dying in a given year approximately doubling every eight years. The model is not intended to describe the elevated mortality of infancy and early childhood, and at the very highest ages estimates of the mortality trajectory are sensitive to data quality and modelling assumptions. Some demographic studies report a slowing of the increase in death rates or a plateau among the oldest-old, while others find that cleaned data remain compatible with Gompertz–Makeham-type behaviour over a wide age range.

Because of its simple functional form and interpretable parameters, the Gompertz–Makeham law is widely used in actuarial science for constructing life tables and pricing life insurance and pension products, in demography and gerontology for modelling adult mortality, and in reliability theory of ageing and longevity as a parametric survival distribution for biological and technical systems.