  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 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 our test statistic,
as well as upper bounds on the type II error for an appropriate class of strict
alternatives. Lastly, we provide an efficient numerical scheme for our test
statistic, by way of an entropic Frank-Wolfe algorithm. Some experiments based
on synthetic data sets illuminates the success of our approach empirically.
