Wasserstein tests for equality of several groups for distributional data

A hypothesis testing framework is introduced for comparing groups of distributional data based on Wasserstein geometry, which measures geometric distances between empirical distributions. Since distributional data reside in a non-Euclidean space, conventional operations such as addition and subtraction do not apply, posing challenges in defining statistical measures like mean and variance. The proposed approach addresses this challenge by estimating intrinsic mean and variance in Wasserstein space and constructing test statistics for group comparison. The method enables meaningful comparisons of non-Euclidean data while preserving their geometric structure. The asymptotic distributions of the test statistics are derived under mild regularity conditions, and their power properties are analyzed. The performance of the method is demonstrated through simulation studies and an application to biomedical image data involving glioblastoma multiforme.