Shift-Dispersion Decompositions of Wasserstein and Cram\'er Distances

TL;DR

This paper introduces a novel decomposition of Wasserstein and Cramér distances into shift and dispersion components, improving interpretability and sensitivity analysis of distribution differences.

Contribution

It provides a new theoretical framework for decomposing these divergences, with unique properties in location-scale families and applications in forecast evaluation and survey design.

Findings

Decompositions satisfy natural properties and are unique in location-scale families.

Allow sensitivity analysis to shifts and dispersions in distributions.

Applied to temperature forecast evaluation and economic survey design.

Abstract

Divergence functions are measures of distance or dissimilarity between probability distributions that serve various purposes in statistics and applications. We propose decompositions of Wasserstein and Cram\'er distances$-$which compare two distributions by integrating over their differences in distribution or quantile functions$-$into directed shift and dispersion components. These components are obtained by dividing the differences between the quantile functions into contributions arising from shift and dispersion, respectively. Our decompositions add information on how the distributions differ in a condensed form and consequently enhance the interpretability of the underlying divergences. We show that our decompositions satisfy a number of natural properties and are unique in doing so in location-scale families. The decompositions allow to derive sensitivities of the divergence measures to changes in location and dispersion, and they give rise to weak stochastic order relations that are linked to the usual stochastic and the dispersive order. Our theoretical developments are illustrated in two applications, where we focus on forecast evaluation of temperature extremes and on the design of probabilistic surveys in economics.