Background: Life expectancy at birth (e0), life years lost at death (e†), and lifetable entropy (H) are key indicators that capture average lifespan and lifespan variation. Expressions and relationships among these summary measures form the basis to analytically derive a range of formal demographic relationships, that build on each other and together help create new insights. Even though many elegant relationships are known, new ones are still to be discovered. Results: The sum of life expectancy and life years lost at death, scaled by the level and rate parameters a and b of the Gompertz mortality model, equals one. This plain relationship has mathematical beauty and connects key demographic measures. It directly implies further relationships, and allows connecting existing ones. It can be interpreted as a pace–shape decomposition of lifespan. Contribution: We contribute a useful relationship to complement analytical tools for studying life expectancy and lifespan variation. It can reveal macro-level regularities that may aid development of novel forecasting methods in the future. It could also support more comparative research across species by quantifying the relative impact of the environment on species’ life histories. We also propose the ratio a/b as a potential metric to signal major trend changes in mortality improvements.

中文翻译：

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寿命和损失的生命年如何等同于统一（作者：Annette Baudisch、Jose Manuel Aburto）

背景：出生时预期寿命 (e0)、死亡时损失生命年 (e†) 和寿命熵 (H) 是反映平均寿命和寿命变化的关键指标。这些汇总指标之间的表达和关系构成了分析得出一系列正式人口统计关系的基础，这些关系相互基础，共同帮助创造新的见解。尽管许多优雅的关系是已知的，但新的关系仍有待发现。结果：按 Gompertz 死亡率模型的水平和比率参数 a 和 b 缩放的预期寿命和死亡损失寿命年的总和等于 1。这种简单的关系具有数学之美，并将关键的人口统计指标联系起来。它直接意味着进一步的关系，并允许连接现有的关系。它可以被解释为寿命的速度-形状分解。贡献：我们提供了有用的关系来补充研究预期寿命和寿命变化的分析工具。它可以揭示宏观层面的规律，有助于未来新预测方法的开发。它还可以通过量化环境对物种生命史的相对影响来支持更多的跨物种比较研究。我们还建议将比率​​ a/b 作为指示死亡率改善的主要趋势变化的潜在指标。