Background: The modal age at death (or mode) is an important indicator of longevity associated with different mortality regularities. Accurate estimates of the mode are essential, but existing methods are not always able to provide them. Objective: Our objective is to develop a method to estimate the modal age at death, which is purely based on its mathematical properties. Methods: The mode maximizes the density of the age-at-death distribution. In addition, at the mode, the rate of aging equals the force of mortality. Using these properties, we develop a novel discrete estimation method for the mode, the discretized derivative tests (DDT) method, and compare its outcomes to those of other existing models. Results: Both the modal age at death and the rate of aging have been increasing since 1960 in low-mortality countries. The DDT method produces close estimates to the ones generated by the P-spline smoothing. Conclusions: The modal age at death plays a central role in estimating longevity advancement, quantifying mortality postponement, and estimating the rate of aging. The novel DDT method proposed here provides a simple and mathematically based estimation of the modal age at death. The method accounts for the mathematical properties of the mode and is not computationally demanding. Contribution: Our research was motivated by James W. Vaupel, who wanted to find a way to accurately estimate the mode based on its mathematical properties. This article also expands on some of his last research papers that link the modal age at death for populations to the one for individuals.

中文翻译：

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模式长寿：一种离散导数检验方法，用于准确估计成人死亡时的模式年龄（Paola Vazquez-Castillo、Marie-Pier Bergeron-Boucher、Trifon Missov）

背景：死亡时的众数年龄（或模式）是与不同死亡规律相关的长寿的重要指标。准确估计众数至关重要，但现有方法并不总是能够提供这些估计。目标：我们的目标是开发一种完全基于其数学特性来估计死亡年龄的方法。方法：该模式最大化死亡年龄分布的密度。此外，在这种模式下，衰老的速度等于死亡的力量。利用这些特性，我们开发了一种新颖的模型离散估计方法，即离散导数测试（DDT）方法，并将其结果与其他现有模型的结果进行比较。结果：自 1960 年以来，低死亡率国家的最态死亡年龄和老龄化率都在上升。 DDT 方法生成的估计值与 P 样条平滑生成的估计值接近。结论：众数死亡年龄在估计寿命提前、量化死亡率延迟和估计衰老速度方面发挥着核心作用。这里提出的新颖的 DDT 方法提供了一种简单且基于数学的死亡模式年龄估计。该方法考虑了模式的数学特性并且计算量不大。贡献：我们的研究是由 James W. Vaupel 发起的，他希望找到一种根据模态的数学特性准确估计模态的方法。本文还扩展了他最近的一些研究论文，这些论文将人口的模态死亡年龄与个人的模态死亡年龄联系起来。