Small Sample Behavior of Wasserstein Projections, Connections to Empirical Likelihood, and Other Applications

The empirical Wasserstein projection (WP) distance quantifies the Wasserstein distance from the empirical distribution to a set of probability measures satisfying given expectation constraints. The WP is a powerful tool because it mitigates the curse of dimensionality inherent in the Wasserstein distance, making it valuable for various tasks, including constructing statistics for hypothesis testing, optimally selecting the ambiguity size in Wasserstein distributionally robust optimization, and studying algorithmic fairness. While the weak convergence analysis of the WP as the sample size $n$ grows is well understood, higher-order (i.e., sharp) asymptotics of WP remain unknown. In this paper, we study the second-order asymptotic expansion and the Edgeworth expansion of WP, both expressed as power series of $n^{-1/2}$. These expansions are essential to develop improved confidence level accuracy and a power expansion analysis for the WP-based tests for moment equations null against local alternative hypotheses. As a by-product, we obtain insightful criteria for comparing the power of the Empirical Likelihood and Hotelling's $T^2$ tests against the WP-based test. This insight provides the first comprehensive guideline for selecting the most powerful local test among WP-based, empirical-likelihood-based, and Hotelling's $T^2$ tests for a null. Furthermore, we introduce Bartlett-type corrections to improve the approximation to WP distance quantiles and, thus, improve the coverage in WP applications.