On the relationship between the Wasserstein distance and differences in life expectancy at birth

TL;DR

This paper demonstrates that, under certain conditions, the Wasserstein distance between two life table distributions equals the difference in life expectancy at birth, providing a new interpretation of this metric in demographic analysis.

Contribution

It establishes a mathematical and empirical link between Wasserstein distance and life expectancy differences, especially when survivorship functions do not cross.

Findings

Wasserstein distance equals life expectancy gap when survivorship functions do not cross.

Empirical validation using Human Mortality Database data.

Provides a new interpretation of distributional differences in mortality.

Abstract

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.