Statistical inference of convex order by Wasserstein projection

TL;DR

This paper introduces a new statistical test for convex order between distributions using Wasserstein projection distance, providing theoretical guarantees and an efficient computational scheme, with promising results on synthetic data.

Contribution

It develops the first statistical test for convex order in multiple dimensions based on Wasserstein projection, with proven stability, consistency, and an efficient algorithm.

Findings

The test accurately detects convex order deviations.

Theoretical bounds on p-value and error rates are established.

Experimental results demonstrate the method's effectiveness on synthetic data.

Abstract

Ranking distributions according to a stochastic order has wide applications in diverse areas. Although stochastic dominance has received much attention, convex order, particularly in general dimensions, has yet to be investigated from a statistical point of view. This article addresses this gap by introducing a simple statistical test for convex order based on the Wasserstein projection distance. This projection distance not only encodes whether two distributions are indeed in convex order, but also quantifies the deviation from the desired convex order and produces an optimal convex order approximation. Lipschitz stability of the backward and forward Wasserstein projection distance is proved, which leads to elegant consistency and concentration results of the estimator we employ as our test statistic. Combining these with state of the art results regarding the convergence rate of empirical distributions, we also derive upper bounds for the $p$-value and type I error of our test statistic, as well as upper bounds on the type II error for an appropriate class of strict alternatives. With proper choices of families of distributions, we further attain that the power of the proposed test increases to one as the number of samples grows to infinity. Lastly, we provide an efficient numerical scheme for our test statistic, by way of an entropic Frank-Wolfe algorithm. Experiments based on synthetic data sets illuminate the success of our approach.