An Exact Poisson Mixture Representation of Siler Remaining Life Expectancy
Abstract
The Siler model describes mortality from birth to old age as the sum of a declining immaturity hazard, a constant background hazard, and an increasing senescence hazard. No standard closed form is available for its remaining life expectancy, which must therefore be evaluated by numerical integration. We show that Siler remaining life expectancy at age x is a Poisson mixture of Gompertz-Makeham remaining life expectancies, that is, a Poisson-weighted average of life expectancies in models with only a constant background hazard and an exponentially increasing senescence hazard. All components share the senescence hazard. Their background hazards increase from one component to the next by the rate at which immaturity mortality declines. The Poisson mean is the immaturity hazard that remains to be accumulated after age x. This representation yields an exact series of generalized exponential integrals with a closed-form bound on the truncation error. Replacing the background hazard by its discounted counterpart also gives the Laplace transform of the remaining lifetime. The mixture bounds the life expectancy lost to immaturity mortality: up to a term below the reciprocal of a2+b1, this loss equals the probability of an immaturity death times the life expectancy with the immaturity hazard removed. In annual Siler fits to Swedish female mortality, we estimate that the loss at birth fell from 4.3 years in 1900 to less than 0.01 years in 2020, which accounts for 15% of the gain in life expectancy at birth over the period.
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BibTeX
@misc{patricio2026exact,
title = {{An Exact Poisson Mixture Representation of Siler Remaining Life Expectancy}},
author = {Silvio C. Patricio},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06376},
eprint = {2611.06376},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06376},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Exact Poisson Mixture Representation of Siler Remaining Life Expectancy
AU - Silvio C. Patricio
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06376
UR - https://arxiv.org/abs/2611.06376
AB - The Siler model describes mortality from birth to old age as the sum of a declining immaturity hazard, a constant background hazard, and an increasing senescence hazard. No standard closed form is available for its remaining life expectancy, which must therefore be evaluated by numerical integration. We show that Siler remaining life expectancy at age x is a Poisson mixture of Gompertz-Makeham remaining life expectancies, that is, a Poisson-weighted average of life expectancies in models with only a constant background hazard and an exponentially increasing senescence hazard. All components share the senescence hazard. Their background hazards increase from one component to the next by the rate at which immaturity mortality declines. The Poisson mean is the immaturity hazard that remains to be accumulated after age x. This representation yields an exact series of generalized exponential integrals with a closed-form bound on the truncation error. Replacing the background hazard by its discounted counterpart also gives the Laplace transform of the remaining lifetime. The mixture bounds the life expectancy lost to immaturity mortality: up to a term below the reciprocal of a_2+b_1, this loss equals the probability of an immaturity death times the life expectancy with the immaturity hazard removed. In annual Siler fits to Swedish female mortality, we estimate that the loss at birth fell from 4.3 years in 1900 to less than 0.01 years in 2020, which accounts for 15% of the gain in life expectancy at birth over the period.
ER -