# An optimal transportation approach for assessing almost stochastic order

## TL;DR

This paper introduces a Wasserstein distance-based index to measure and test for almost stochastic dominance by optimally trimming distributions, providing statistical guarantees and demonstrating good performance through simulations.

## Contribution

It proposes a novel Wasserstein distance approach for assessing almost stochastic order, including asymptotic tests and a new index of disagreement.

## Key findings

- The index effectively measures almost stochastic dominance.
- Asymptotic tests provide statistical guarantees.
- Simulation shows good performance under normal models.

## Abstract

When stochastic dominance $F\leq_{st}G$ does not hold, we can improve agreement to stochastic order by suitably trimming both distributions. In this work we consider the $L_2-$Wasserstein distance, $\mathcal W_2$, to stochastic order of these trimmed versions. Our characterization for that distance naturally leads to consider a $\mathcal W_2$-based index of disagreement with stochastic order, $\varepsilon_{\mathcal W_2}(F,G)$. We provide asymptotic results allowing to test $H_0: \varepsilon_{\mathcal W_2}(F,G)\geq \varepsilon_0$ vs $H_a: \varepsilon_{\mathcal W_2}(F,G)<\varepsilon_0$, that, under rejection, would give statistical guarantee of almost stochastic dominance. We include a simulation study showing a good performance of the index under the normal model.