BACKGROUND
Previous research has derived bounds on the remaining life expectancy function e(x) that connect survivorship and remaining life expectancy at two age values and therefore can be used to, among other things, estimate life expectancy at birth when the population’s full mortality trajectory is not known.

RESULTS
We show that the aforementioned bounds emerge from using particular two-node closed quadrature rules and prove a theorem that establishes conditions for when an n-node closed rule respects those bounds for e(x). This enables the usage of known high-accuracy rules that stay within the bounds and provide new high-accuracy estimates for e(x). We show that among this set of rules are ones that yield exact estimates for e(x). We illustrate our work empirically using life table data from French females since 1816 and discover a new empirical regularity linking life expectancy at birth in the data set to survivorship and remaining life expectancy at age 20.



中文翻译：

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通过基于正交规则的方法改进剩余预期寿命的界限和高精度估计（Oscar Fernandez）

背景
先前的研究已经推导出剩余预期寿命函数 e(x) 的界限，该函数将两个年龄值的生存率和剩余预期寿命联系起来，因此除其他外，可用于在人口的完整死亡率轨迹为未知。

结果
我们证明了上述边界是从使用特定的双节点封闭正交规则中产生的，并证明了一个定理，该定理为 n 节点封闭规则何时遵守 e(x) 的边界建立了条件。这使得能够使用保持在边界内的已知高精度规则，并为 e(x) 提供新的高精度估计。我们表明，在这组规则中，有一些规则可以产生 e(x) 的精确估计。我们使用 1816 年以来法国女性的生命表数据从经验上说明了我们的工作，并发现了一种新的经验规律，将数据集中出生时的预期寿命与 20 岁时的存活率和剩余预期寿命联系起来。