The Wasserstein distance is a metric for assessing distributional differences. The measure originates in optimal transport theory and can be interpreted as the minimal cost of transforming one distribution into another. In this paper, the Wasserstein distance is applied to life table age-at-death distributions. The main finding is that, under certain conditions, the Wasserstein distance between two age-at-death distributions equals the corresponding gap in life expectancy at birth ($e_0$). More specifically, the paper shows mathematically and empirically that this equivalence holds whenever the survivorship functions do not cross. For example, this applies when comparing mortality between women and men from 1990 to 2020 using data from the Human Mortality Database. In such cases, the gap in $e_0$ reflects not only a difference in mean ages at death but can also be interpreted directly as a measure of distributional difference.

翻译：Wasserstein距离是一种用于评估分布差异的度量指标。该度量起源于最优传输理论，可解释为将一个分布转换为另一个分布的最小成本。本文中，Wasserstein距离被应用于生命表死亡年龄分布的分析。主要研究发现是，在特定条件下，两个死亡年龄分布之间的Wasserstein距离等于对应的出生时预期寿命（$e_0$）差异。具体而言，本文通过数学证明和实证分析表明，当生存函数不存在交叉时，这种等价关系始终成立。例如，使用人类死亡率数据库1990年至2020年的数据比较女性和男性死亡率时即符合此条件。在此类情况下，$e_0$差异不仅反映了平均死亡年龄的差别，还可直接解释为分布差异的度量指标。