Background: Indicators of longevity like the life expectancy at birth or the modal age at death are always positively affected by improvements in mortality. Instead, for lifespan variation it has been shown that there exists a threshold age above and below which averting deaths respectively increases or decreases such variation. Objective: Within a Gompertz force of mortality setting, we aim to provide approximations of the life expectancy at birth and the threshold age of the life table entropy in terms of the modal age at death, highlighting the interrelationships holding among the three. Results: In the Gompertz framework, a tight relationship exists between the life expectancy at birth, the threshold age of the life table entropy, and the modal age at death, with the former two moving together and in parallel to the latter. We apply this theoretical result to life table data from the Human Mortality Database to show how the different relationships evolve over time. We observe a remarkable association between the modal and the threshold ages, even in populations with high mortality levels. Contribution: We provide approximations of the life expectancy at birth and the threshold age of the life table entropy in terms of the Gompertz modal age at death. This is a mathematical demography paper that builds upon previous research by James W. Vaupel and illustrates the beauty – and oftentimes simplicity – of the mathematical relationships between demographic concepts.

中文翻译：

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关于预期寿命、死亡模态年龄和生命表熵的阈值年龄之间的关系（作者：Chiara Micheletti、Francisco Villavicencio）


背景：长寿指标，如出生时的预期寿命或死亡时的模态年龄，总是受到死亡率改善的积极影响。相反，对于寿命变化，已经表明存在一个阈值年龄，高于和低于该阈值年龄避免死亡分别会增加或减少这种变化。目的：在 Gompertz 死亡力设置中，我们的目标是根据死亡模态年龄提供出生时预期寿命和生命表熵的阈值年龄的近似值，突出三者之间的相互关系。结果： 在 Gompertz 框架中，出生时的预期寿命、生命表熵的阈值年龄和死亡时的模态年龄之间存在紧密关系，前两者一起移动并与后者平行。我们将这一理论结果应用于人类死亡率数据库中的生命表数据，以展示不同的关系如何随着时间的推移而演变。我们观察到模态年龄和阈值年龄之间存在显着关联，即使在死亡率水平高的人群中也是如此。贡献：我们根据死亡时的 Gompertz 模态年龄提供了出生时预期寿命和生命表熵的阈值年龄的近似值。这是一篇数学人口学论文，建立在 James W. Vaupel 以前的研究之上，说明了人口学概念之间数学关系的美感——而且往往是简单的。