Multivariate Symmetry: Distribution-Free Testing via Optimal Transport
Speaker: Gonzalo Mena
Abstract: Sign and rank statistics have been widely studied because they enable flexible and powerful non-parametric testing on a number of problems. I will focus on testing for symmetry. In the one-dimensional case, the null hypothesis is H0: X=-X, for which the so-called sign test and Wilcoxon's absolute sum-rank tests are known to enjoy distribution-freeness under the null and remarkable asymptotic efficiency properties, regardless of their non-parametric nature. Unfortunately, it is not fully clear how the notion of rank and sign can be extended to the multidimensional case. In this discussion, I will present the work by Huang and Sen (2023). I will comment on different notions of symmetry that appear in the multidimensional case and how the mathematical framework of optimal transportation can be leveraged to develop natural extensions of the sign and rank statistics, which, in turn, are used as natural statistics for testing for symmetry in the multidimensional case. Remarkably, the nice asymptotic properties of such tests are retained in the multidimensional case, suggesting that the proposed framework adds up to a natural extension. The discussion will consist of an introductory commentary on the efficiency of sign-rank tests in the one-dimensional case (mostly from Van der Vaart's book) and elementary results on optimal transportation, followed by an exposition of the main results in Huang and Sen. Huang, Zhen, and Bodhisattva Sen. "Multivariate symmetry: Distribution-free testing via optimal transport." arXiv preprint arXiv:2305.01839 (2023). https://arxiv.org/pdf/2305.01839.pdf