Wasserstein Projection Tests: Higher-Order Asymptotics and Connections to Empirical Likelihood

Understand

Tests of moment restrictions can be constructed by projecting the empirical distribution onto the set of laws satisfying the null.

• Wasserstein projection (WP) performs this projection by transporting mass in the sample space, rather than only reweighting observed atoms as in empirical likelihood (EL), and therefore incorporates both the ground geometry and the local variation of the moment function.
• Existing WP theory provides a first-order null limit but does not quantify finite-sample calibration or distinguish tests with the same limiting local power.
• We derive stochastic expansions of the rescaled WP statistic under the null and under Pitman-type local alternatives, identifying its order-$n^{-1/2}$ correction with a uniform remainder of order $\widetilde{O}_p(n^{-1})$.