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	${\displaystyle 1-\exp \left[-\lambda x-\frac{\alpha }{\beta }\left(e^{\beta x}-1\right)\right]}$

TheGompertz–Makeham law states that the human death rate is the sum of an age-dependent component (theGompertz function, named afterBenjamin Gompertz),^[1] whichincreases exponentially with age,^[2] and an age-independent component (the Makeham term, named afterWilliam Makeham).^[3] In a protected environment where external causes of death are rare (laboratory conditions, low mortality countries, etc.), the age-independent mortality component is often negligible. In this case the formula simplifies to a Gompertz law of mortality. In 1825, Benjamin Gompertz proposed an exponential increase in death rates with age.

The Gompertz–Makeham law of mortality describes the age dynamics of human mortality rather accurately in the age window from about 30 to 80 years of age. At more advanced ages, some studies have found that death rates increase more slowly – a phenomenon known as thelate-life mortality deceleration^[2] – but more recent studies disagree.^[4]

Percentage surviving to certain ages for males and females in the US in 2019^[5] and in Japan in 2020^[6]

The decline in the humanmortality rate before the 1950s was mostly due to a decrease in the age-independent (Makeham) mortality component, while the age-dependent (Gompertz) mortality component was surprisingly stable.^[2][7] Since the 1950s, a new mortality trend has started in the form of an unexpected decline in mortality rates at advanced ages and "rectangularization" of the survival curve.^[8][9]

It does not model the component that makesinfant mortality higher than regular mortality.

The empirical magnitude of the beta-parameter is about .085, implying a doubling of mortality every .69/.085 = 8 years (Denmark, 2006).

Thehazard function for the Gompertz-Makeham distribution is most often characterised as${\displaystyle h(x)=\alpha e^{\beta x}+\lambda }$.

Thequantile function can be expressed in aclosed-form expression using theLambert W function:^[10]

${\displaystyle Q(u)=\frac{\alpha }{\beta \lambda }-\frac{1}{\lambda }\ln (1-u)-\frac{1}{\beta }{W}_0\left[\frac{\alpha e^{\alpha /\lambda }(1-u{)}^{-(\beta /\lambda )}}{\lambda }\right]}$

The Gompertz law is the same as aFisher–Tippett distribution for the negative of age, restricted to negative values for therandom variable (positive values for age).

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