Makeham Mortality Models as Mixtures

Mortality modeling is crucial to understanding the complex nature of population aging and projecting future trends. The Makeham term is a commonly used constant additive hazard in mortality modeling to capture background mortality unrelated to aging. In this manuscript, we propose representing Makeham mortality models as mixtures that describe lifetimes in a competing-risk framework: an individual dies either according to a baseline mortality mechanism or an exponential distribution, whatever strikes first. The baseline can describe mortality at all ages or just mortality due to aging. By using this approach, we can estimate the share of non-senescent mortality at each adult age, which is an essential contribution to the study of premature and senescent mortality. Our results allow for a better understanding of the underlying mechanisms of mortality and provide a more accurate picture of mortality dynamics in populations.

[1]  T. Missov,et al.  Using a penalized likelihood to detect mortality deceleration , 2023, PloS one.

[2]  Artur J. Lemonte,et al.  On the Gompertz–Makeham law: A useful mortality model to deal with human mortality , 2022, Brazilian Journal of Probability and Statistics.

[3]  SPECIAL FUNCTIONS , 2021, Water‐Quality Engineering in Natural Systems.

[4]  Artur J. Lemonte,et al.  On closed-form expressions to Gompertz-Makeham life expectancy. , 2020, Theoretical population biology.

[5]  Marcelo Pereyra,et al.  Revisiting Maximum-A-Posteriori Estimation in Log-Concave Models , 2016, SIAM J. Imaging Sci..

[6]  T. Missov,et al.  Sensitivity of model-based human mortality measures to exclusion of the Makeham or the frailty parameter , 2016 .

[7]  J. Vaupel,et al.  The Gompertz force of mortality in terms of the modal age at death , 2015 .

[8]  P. Jodrá.  On Order Statistics from the Gompertz–Makeham Distribution and the Lambert W Function , 2013 .

[9]  John K Kruschke,et al.  Bayesian data analysis. , 2010, Wiley interdisciplinary reviews. Cognitive science.

[10]  P. Deb.  Finite Mixture Models , 2008 .

[11]  W Siler,et al.  Parameters of mortality in human populations with widely varying life spans. , 2007, Statistics in medicine.

[12]  M. Plummer,et al.  CODA: convergence diagnosis and output analysis for MCMC , 2006 .

[13]  Zeljko Bajzer,et al.  Combining Gompertzian growth and cell population dynamics. , 2003, Mathematical biosciences.

[14]  S. Jay Olshansky,et al.  Ever since gompertz , 1997, Demography.

[15]  D. Brillinger,et al.  The natural variability of vital rates and associated statistics. , 1986, Biometrics.

[16]  B. Dhillon.  Life Distributions , 1981, IEEE Transactions on Reliability.

[17]  J. Vaupel,et al.  The impact of heterogeneity in individual frailty on the dynamics of mortality , 1979, Demography.

[18]  W. Siler.  A Competing‐Risk Model for Animal Mortality , 1979 .

[19]  W. M. Makeham.  On the Further Development of Gompertz's Law , 1889 .

[20]  T. N. Thiele.  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 , 1871 .

[21]  Dmitri A. Jdanov,et al.  Human Mortality Database , 2019, Encyclopedia of Gerontology and Population Aging.

[22]  David P. Smith,et al.  On the Law of Mortality , 2013 .

[23]  F. Norström.  The Gompertz-Makeham distribution , 1997 .

[24]  J. Wilmoth,et al.  Development of Oldest-Old Mortality, 1950-1990: Evidence from 28 Developed Countries , 1996 .

[25]  W. M. Makeham,et al.  On the Law of Mortality and the Construction of Annuity Tables , 1860 .

[26]  Benjamin Gompertz,et al.  XXIV. 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. In a letter to Francis Baily, Esq. F. R. S. &c , 1825, Philosophical Transactions of the Royal Society of London.