MC: Mortality curve fitting

In demofit: Parametric Mortality Curve Fitting and Mortality Forecasting Tools

Mortality curve fitting

Description

Fits parametric mortality curves to observed mortality rates using multiple optimisation strategies.

Usage

MC(
  x,
  m,
  curve = c("gompertz", "makeham", "oppermann", "thiele", "wittsteinbumsted", "perks",
    "weibull", "vandermaen", "beard", "heligmanpollard", "rogersplanck", "siler",
    "martinelle", "thatcher", "gompertz2", "makeham2", "oppermann2", "thiele2",
    "wittsteinbumsted2", "perks2", "weibull2", "vandermaen2", "beard2",
    "heligmanpollard2", "rogersplanck2", "siler2", "martinelle2", "thatcher2"),
  w = rep(1, length(x))
)

Arguments

x	vector of ages.
m	vector of mortality rates.
curve	name of mortality curve (including gompertz, makeham, oppermann, thiele, wittsteinbumsted, perks, weibull, vandermaen, beard, heligmanpollard, rogersplanck, siler, martinelle, thatcher, gompertz2, makeham2, oppermann2, thiele2, wittsteinbumsted2, perks2, weibull2, vandermaen2, beard2, heligmanpollard2, rogersplanck2, siler2, martinelle2, thatcher2, where first 14 curves' parameters are unconstrained and last 14 curves' parameters are generally restricted to be positive).
w	vector of weights (default = 1).

Details

User can choose one of the following 14 mortality curves (with their suitable age ranges in brackets):

• Gompertz (1825) (ages 30-90) - m_x = B e^{Cx}

• Makeham (1860) (ages 20-90) - m_x = A + B e^{Cx}

• Oppermann (1870) (ages 0-20) - m_x = \frac{A}{\sqrt{x+1}} + B + C\sqrt{x+1}

• Thiele (1871) (ages 0-90) - m_x = A_1 e^{-B_1 x} + A_2 e^{-0.5 B_2 (x-C)^2} + A_3 e^{B_3 x}

• Wittstein & Bumsted (1883) (ages 0-90) - m_x = \frac{A^{-(Bx)^N}}{B} + A^{-(M-x)^N}

• Perks (1932) (ages 20-100+) - m_x = \frac{A+BC^x}{1+DC^x}

• Weibull (1939) (ages 50-100+) - m_x = Bx^C

• Van der Maen (1943) (ages 80-100+) - m_x = A + Bx + Cx^2 + \frac{I}{N-x}

• Beard (1971) (ages 50-100+) - m_x = \frac{A e^{Cx}}{1+B e^{Cx}}

• Heligman & Pollard (1980) (ages 0-100+) - m_x = A^{(x+B)^C} + D e^{-E(ln(x)-ln(F))^2} + \frac{GH^x}{1+GH^x}

• Rogers & Planck (1983) (ages 0-100+) - m_x = A_0 + A_1 e^{-Ax} + A_2 e^{-B(x-U)-e^{-C(x-U)}} + A_3 e^{Dx}

• Siler (1983) (ages 0-90) - m_x = A_1 e^{-B_1 x} + A_2 + A_3 e^{B_3 x}

• Martinelle (1987) (ages 20-100+) - m_x = \frac{A+B e^{Cx}}{1+D e^{Cx}} + E e^{Cx}

• Thatcher (1999) (ages 20-100+) - m_x = A + \frac{B e^{Cx}}{1+D e^{Cx}}

The mortality curves are fitted by employing multiple optimisation strategies simultaneously (including PORT routines, Nelder-Mead method, and Levenberg-Marquardt algorithm) and selecting one with smallest weighted least squares on ln(m_x). Constrained parameterisations offer traditional interpretability while unconstrained parameterisations provide increased flexibility.

Value

An object of class based on selected mortality curve with associated S3 methods coef, fitted, predict, plot, deviance, and residuals.

References

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 of London, 115(1825), 513-583.

Makeham, W.M. (1860). On the law of mortality and the construction of annuity tables. Journal of the Institute of Actuaries, 8(6), 301-310.

Oppermann, L.H.F. (1870). On the graduation of life tables, with special application to the rate of mortality in infancy and childhood. The Institute of Actuaries.

Thiele, T.N. (1871). On a mathematical formula to express the rate of mortality throughout the whole of life, tested by a series of observations made use of by the Danish Life Insurance Company of 1871. Journal of the Institute of Actuaries and Assurance Magazine, 16(5), 313-329.

Wittstein, T. and Bumsted, D.A. (1883). The mathematical law of mortality. Journal of the Institute of Actuaries and Assurance Magazine, 24(3), 153-173.

Perks, W. (1932). On some experiments in the graduation of mortality statistics. Journal of the Institute of Actuaries, 63(1), 12-57.

Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of Applied Mechanics, 18(3), 293-297.

Beard, R.E. (1971). Some aspects of theories of mortality, cause of death analysis, forecasting and stochastic processes. Biological Aspects of Demography, 57-68.

Heligman, L. and Pollard, J.H. (1980). The age pattern of mortality. Journal of the Institute of Actuaries, 107(1), 49-80.

Rogers, A. and Planck, F. (1983). Model: A general program for estimating parametrized model schedules of fertility, mortality, migration, and marital and labor force status transitions. IIASA Working Paper WP-83-102.

Siler, W. (1983). Parameters of mortality in human populations with widely varying life spans. Statistics in Medicine, 2(3), 373-380.

Martinelle, S. (1987). A generalized Perks formula for old-age mortality. Statistiska Centralbyran.

Thatcher, A.R. (1999). The long-term pattern of adult mortality and the highest attained age. Journal of the Royal Statistical Society Series A, 162(1), 5-43.

Tabeau, E. (2001). A review of demographic forecasting models for mortality. Forecasting Mortality in Developed Countries. European Studies of Population, Vol 9.

Examples

x <- 60:89
set.seed(123); m <- 0.0000082*exp(0.10771*c(60:89)+rnorm(30,0,0.1))
fit <- MC(x=x,m=m,curve="gompertz")
coef(fit)
fitted(fit)
predict(fit,62.5)
plot(fit)
deviance(fit)
residuals(fit)

demofit documentation built on April 9, 2026, 5:09 p.m.