Multidimensional Stochastic Dominance Test Based on Center-outward Quantiles

TL;DR

This paper introduces a new multivariate stochastic dominance test based on center-outward quantiles, utilizing optimal transport for computational efficiency and providing rigorous statistical inference methods.

Contribution

It develops novel multivariate stochastic dominance concepts using center-outward quantiles and proposes entropy-regularized optimal transport for efficient estimation and testing.

Findings

The proposed tests perform well in finite samples.

Bootstrap methods validate the test procedures.

The approach is computationally feasible and theoretically rigorous.

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 Cram\'er- 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.