Warsserstein Tests for Equality of Several Groups for Distributional Data

Abstract The study aims to develop a Wasserstein test for the equality of groups for distri-butional data using Wasserstein geometry to account for the geometric distance between empirical distributions. Non-Euclidean data, such as distributional data, do not exist in a linear vector space; therefore, subtraction or addition may not have a meaningful operator. Furthermore, defining measures such as the mean and variance can also be challenging for this problem. This framework enables us to estimate the intrinsic mean and variance using the Wasserstein metric in the Wasserstein space and then to compute the test statistics that can capture the group difference. Therefore, we can compare the population of non-Euclidean data as well as handle the non-Euclidean data. We also derive the asymptotic distribution of the test statistics under mild regularity conditions and establish the asymptotic properties for power performance. We demonstrated the advantages of our proposed approach through simulation studies on various special cases, as well as through its application to biomedical image data, specifically focusing on glioblastoma multiforme.