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距离是评估分布差异的一种度量。该度量源于最优传输理论，可解释为将一个分布变换为另一个分布的最小成本。本文将Wasserstein距离应用于生命表年龄死亡分布。主要发现是，在一定条件下，两个年龄死亡分布之间的Wasserstein距离等于相应的出生时预期寿命差距（$e_0$）。更具体地说，本文从数学和实证角度证明，当生存函数不相交时，这种等价关系成立。例如，在比较1990年至2020年人类死亡率数据库中女性和男性死亡率时即适用此情况。此时，$e_0$的差距不仅反映了死亡平均年龄的差异，还可直接解释为分布差异的度量。