Multidimensional Stochastic Dominance Test Based on Center-outward Quantiles

Abstract

Stochastic dominance (SD) provides a quantile-based partial ordering of random variables and has broad applications. Its extension to multivariate settings, however, is challenging due to the lack of a canonical ordering in ${\mathbb{R}}^d$ ($d\ge 2$) and the set-valued character of multivariate quantiles. Based on the multivariate center-outward quantile function in Hallin et al. (2021), this paper proposes new first- and second-order multivariate stochastic dominance (MSD) concepts through comparing contribution functions defined over quantile contours and regions. To address computational and inferential challenges, we incorporate entropy-regularized optimal transport, which ensures faster convergence rate and tractable estimation. We further develop consistent Kolmogorov-Smirnov and Cramér- von Mises type test statistics for MSD, establish bootstrap validity, and demonstrate through extensive simulations good finite-sample performance of the tests. Our approach offers a theoretically rigorous, and computationally feasible solution for comparing multivariate distributions.