Gompertz–Makeham law of mortality

Gompertz–Makeham
Parameters	$\alpha >0$ (real) $\beta >0$ (real) $\lambda >0$ (real)
Support	$x\in \mathbb {R} ^{+}$
PDF	$(\alpha e^{\beta x}+\lambda )\cdot \exp(-\lambda x-{\frac {\alpha }{\beta }}(e^{\beta x}-1))$
CDF	$1-\exp(-\lambda x-{\frac {\alpha }{\beta }}(e^{\beta x}-1))$

The Gompertz–Makeham law states that the human death rate is the sum of an age-independent component (the Makeham term, named after William Makeham)^[1] and an age-dependent component (the Gompertz function, named after Benjamin Gompertz),^[2] which increases exponentially with age.^[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 the late-life mortality deceleration^[3] – but more recent studies disagree.^[4]

Estimated probability of a person dying at each age, for the U.S. in 2003 . Mortality rates increase exponentially with age after age 30.

The decline in the human mortality 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.^[3]^[5] 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.^[6]^[7]

The hazard function for the Gompertz-Makeham distribution is most often characterised as $h(x)=\alpha e^{\beta x}+\lambda$ . The empirical magnitude of the beta-parameter is about .085, implying a doubling of mortality every .69/.085 = 8 years (Denmark, 2006).

The quantile function can be expressed in a closed-form expressions using the Lambert W function:^[8]

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 a Fisher–Tippett distribution for the negative of age, restricted to negative values for the random variable (positive values for age).

References

↑ Makeham, W. M. (1860). "On the Law of Mortality and the Construction of Annuity Tables". J. Inst. Actuaries and Assur. Mag. 8: 301–310.
↑ Gompertz, B. (1825). "On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies". Philosophical Transactions of the Royal Society. 115: 513–585. doi:10.1098/rstl.1825.0026.
1 2 3 Leonid A. Gavrilov & Natalia S. Gavrilova (1991) The Biology of Life Span: A Quantitative Approach. New York: Harwood Academic Publisher, ISBN 3-7186-4983-7
↑ Gavrilov, Leonid A.; Gavrilova, Natalia S. (2011). "Mortality Measurement at Advanced Ages: A Study of the Social Security Administration Death Master File" (PDF). North American Actuarial Journal: 432–447.
↑ Gavrilov, L. A.; Gavrilova, N. S.; Nosov, V. N. (1983). "Human life span stopped increasing: Why?". Gerontology. 29 (3): 176–180. doi:10.1159/000213111.
↑ Gavrilov, L. A.; Nosov, V. N. (1985). "A new trend in human mortality decline: derectangularization of the survival curve". Age. 8 (3): 93.
↑ Gavrilova, N. S.; Gavrilov, L. A. (2011). "Stárnutí a dlouhovekost: Zákony a prognózy úmrtnosti pro stárnoucí populace" [Ageing and Longevity: Mortality Laws and Mortality Forecasts for Ageing Populations]. Demografie (in Czech). 53 (2): 109–128.
↑ Jodrá, P. (2009). "A closed-form expression for the quantile function of the Gompertz–Makeham distribution". Mathematics and Computers in Simulation. 79 (10): 3069–3075. doi:10.1016/j.matcom.2009.02.002.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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Gompertz–Makeham law of mortality

See also

References