Optimal Transport-Based Inference for Distributional Data

Abstract

The analysis of distributional data where each observational unit is itself a probability distribution has become increasingly important across economics, environmental science, biomedical engineering, and machine learning. Classical multivariate methods are fundamentally inadequate for such data because the space of probability measures lacks the linear structure assumed by traditional techniques. Optimal transport (OT) theory, and in particular the 2-Wasserstein metric, provides a geometrically coherent framework for statistical inference on probability distributions. This paper develops a comprehensive methodology for OT-based inference on distributional data, encompassing one-sample, two-sample, and multi-group hypothesis tests formulated in Wasserstein space. We establish the asymptotic theory underpinning these tests, including central limit theorems for Fréchet means and variance statistics. Entropic regularization via the Sinkhorn algorithm is incorporated to render the methodology computationally tractable. We evaluate our framework through extensive Monte Carlo simulations and two empirical applications: income distribution analysis across economic regions and the comparison of precipitation distributions in climate science. Our results demonstrate that Wasserstein-based tests offer substantially greater power than classical alternatives when distributional shape differences are present, while maintaining nominal size under the null hypothesis.