When stochastic dominance F≤stG does not hold, we can improve
agreement to stochastic order by suitably trimming both distributions. In this
work we consider the L2−Wasserstein distance, W2, to stochastic
order of these trimmed versions. Our characterization for that distance
naturally leads to consider a W2-based index of disagreement with
stochastic order, εW2(F,G). We provide asymptotic
results allowing to test H0:εW2(F,G)≥ε0 vs Ha:εW2(F,G)<ε0, 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