Missov and Lenart (2013) derived closed-form solutions to the life expectancy and remaining life expectancy at age x when the mortality is governed by a Gompertz-Makeham hazard, which is a parametric model commonly applied to human mortality data at adult and old ages. However, the closed-form expressions provided by these authors are not correct. We provide, therefore, valid and correct expressions using elementary calculus. We also consider numerical studies using real data to show that the formulas we provide deliver satisfactory results.

中文翻译：

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以封闭形式表达Gompertz-Makeham的预期寿命。

当死亡率由Gompertz-Makeham危害控制时，Missov和Lenart（2013）得出x年龄时的预期寿命和剩余预期寿命的封闭式解，这是一个参数模型，通常用于成人和老年人的死亡率数据。但是，这些作者提供的封闭形式的表达是不正确的。因此，我们使用基本演算来提供有效且正确的表达式。我们还考虑使用实际数据进行数值研究，以表明我们提供的公式可提供令人满意的结果。