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On closed-form expressions to Gompertz-Makeham life expectancy.
Fredy Castellares1, Silvio C Patrício1, Artur J Lemonte2
1Departamento de Estatística, Universidade Federal de Minas Gerais, Belo Horizonte/MG, Brazil.
Theoretical Population Biology
|May 15, 2020
Summary
This study corrects errors in previous life expectancy calculations for the Gompertz-Makeham hazard model. New, accurate formulas are presented and validated with real-world mortality data.
Area of Science:
- Actuarial science
- Demography
- Mathematical statistics
Background:
- The Gompertz-Makeham hazard model is widely used for human mortality data.
- Previous closed-form solutions for life expectancy under this model were found to be incorrect.
- Accurate life expectancy calculations are crucial for actuarial and demographic studies.
Purpose of the Study:
- To derive correct closed-form solutions for life expectancy and remaining life expectancy at age x.
- To address and rectify inaccuracies in existing formulas derived by Missov and Lenart (2013).
- To validate the newly derived formulas using empirical mortality data.
Main Methods:
- Application of elementary calculus to derive new mathematical expressions.
- Analysis of human mortality data.
- Numerical studies to test the accuracy and applicability of the derived formulas.
Main Results:
- Corrected closed-form expressions for life expectancy and remaining life expectancy were successfully derived.
- The new formulas provide accurate results when applied to real mortality data.
- The findings demonstrate the reliability of the corrected expressions for actuarial applications.
Conclusions:
- The previously published closed-form solutions for life expectancy under the Gompertz-Makeham hazard model were indeed incorrect.
- The newly derived formulas offer a valid and accurate alternative for calculating life expectancy.
- The validated formulas are suitable for practical use in actuarial science and demographic research.
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