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

This paper establishes higher-order asymptotic theory for Wasserstein Projection tests, deriving Edgeworth expansions and Bartlett corrections to improve finite-sample calibration and power, while demonstrating their superior performance over Empirical Likelihood and Hotelling's $T^2$ in regimes governed by derivative-based curvature through both theoretical analysis and numerical experiments.

Original authors: Sirui Lin, Jose Blanchet, Peter Glynn, Viet Anh Nguyen

Published 2026-08-18

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CC BY 4.0

Original authors: Sirui Lin, Jose Blanchet, Peter Glynn, Viet Anh Nguyen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of statistics, researchers often face a fundamental question: does a set of data follow a specific rule, or does it deviate from it? Imagine a factory producing bolts that must be exactly the right length. If the bolts are too long or too short, the machine needs adjustment. Statisticians use "moment restrictions" to define these rules mathematically, checking if the average behavior of a dataset matches a theoretical expectation. For decades, the standard way to check this has been to simply reweight the data points we already have, giving more importance to some and less to others, until the average fits the rule. This method, known as empirical likelihood, is efficient but rigid; it treats the data points as fixed islands that can only be moved by changing their weight, not by moving them in space.

A newer approach, called Wasserstein projection, offers a different perspective. Instead of just changing weights, it imagines physically moving the data points through space to satisfy the rule. This movement is measured by a cost function that accounts for the geometry of the space itself—how far apart points are and how the rules change as you move through that space. While this geometric approach has been shown to work well in the long run as data sets grow infinitely large, a critical gap remained: how well does it work with the finite, imperfect data sets we actually have in the real world? Does the extra effort of moving data points through space actually provide a better test than the traditional weighting method when sample sizes are limited?

A team of researchers from Stanford University and the Chinese University of Hong Kong has now filled this gap with a rigorous mathematical analysis. They developed a detailed theory that describes exactly how the Wasserstein projection test behaves in finite samples, going beyond the standard approximations used for decades. Their work reveals that while the traditional weighting method and the new geometric method look identical when viewed from a distance, they diverge significantly when examined closely. The researchers found that the geometric method's performance is governed by the curvature of the rules being tested—how sharply the rules bend in the data space—whereas the traditional method depends on the skewness, or the lopsidedness, of the data values themselves.

The study demonstrates that for certain types of data and rules, the geometric approach offers a distinct advantage in detecting subtle deviations. Specifically, when the rules involve smooth, curved relationships, the method that moves data points through space can detect problems that the weighting method might miss, or at least detect them with greater reliability. The researchers proved that this advantage is not just a lucky fluke but a predictable feature determined by the mathematical shape of the problem. They also showed that the traditional method can outperform the geometric one in other scenarios, particularly when the data values themselves are highly skewed, proving that neither method is universally superior.

To make these findings useful for real-world applications, the team also tackled the computational challenge. Calculating the exact cost of moving data points to satisfy a rule can be incredibly difficult, especially when the rules are complex and non-linear. The researchers designed a new, certified algorithm that solves this problem efficiently. This algorithm does not just guess; it provides a mathematical guarantee that the solution is accurate enough to make a correct decision. They tested this system in a fairness experiment, checking whether a scoring system treated two different groups of people equally. In this test, the geometric method, using a specific way of measuring distance, successfully identified unfairness that the other methods struggled to see, confirming the theoretical predictions about its superior power in curved, geometric settings.

The researchers also provided tools to correct the statistical tests, making them even more accurate for smaller data sets. By applying these corrections, the tests can achieve a level of precision that was previously thought difficult to reach without massive amounts of data. This work bridges the gap between abstract mathematical theory and practical application, showing that the geometry of data matters. It confirms that when we treat data as points in a landscape rather than just a list of numbers, we can build better, more sensitive tools for detecting whether the world is behaving as we expect it to. The findings suggest that for many modern problems involving complex, non-linear rules, the extra computational effort of the geometric approach is well worth the gain in accuracy and reliability.

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

1. Problem Formulation

The paper addresses the construction and analysis of hypothesis tests for moment restrictions. Given an empirical distribution $Q_n$ derived from i.i.d. observations $X_1, \dots, X_n$ and a null model $\mathcal{P}_0 = \{P : E_P[h(X)] = 0\}$ defined by a moment function $h: \mathbb{R}^{d_X} \to \mathbb{R}^{d_h}$ , the goal is to test $H_0: P_\star \in \mathcal{P}_0$ versus $H_1: P_\star \notin \mathcal{P}_0$ .

The authors focus on the Wasserstein Projection (WP) test. Unlike Empirical Likelihood (EL), which projects the empirical law onto $\mathcal{P}_0$ by reweighting observed atoms, WP performs this projection by transporting mass within the sample space. The test statistic is defined as the minimum transportation cost required to align the empirical distribution with the null moment restriction:
$R_n(h) = \inf_{P \in \mathcal{P}_0} W_c(Q_n, P)$
where $W_c$ is the optimal transport cost with a squared Mahalanobis ground cost $c(\bar{x}, x) = \|\bar{x} - x\|_\Sigma^2$ . The test rejects $H_0$ for large values of the rescaled statistic $n R_n(h)$ .

While existing theory establishes that $n R_n(h)$ converges to a quadratic form under the null (providing a first-order asymptotically valid test), this first-order limit fails to:

Quantify finite-sample size distortion (coverage error).
Distinguish the power of WP from competitors (EL and Hotelling's $T^2$ ) when they share the same limiting local power function (specifically in the scalar moment case, $d_h=1$ ).

2. Methodology

The paper develops a higher-order asymptotic analysis to address these gaps. The methodology relies on deriving stochastic expansions of the WP statistic under both the null hypothesis and Pitman-type local alternatives.

Key Technical Steps:

Dual Representation: The authors utilize an exact rescaled dual representation of the projection distance. The statistic is expressed as a supremum over a dual variable $\zeta$ :
$n R_n(h) = \sup_{\zeta \in \mathbb{R}^{d_h}} \left\{ -\zeta^\top H_n - M_n(\zeta) \right\}$
where $H_n$ is the scaled empirical moment and $M_n(\zeta)$ is a convex function involving the ground cost and the moment function.
Stochastic Expansion: By expanding $M_n(\zeta)$ around the origin using Taylor series and controlling the remainder terms via high-probability bounds on empirical quantities (Jacobians, Hessians, and moment envelopes), the authors derive an expansion of the statistic in powers of $n^{-1/2}$ .
Edgeworth Theory: The expansion is translated into Edgeworth expansions for the distribution of the statistic. This allows for the quantification of size errors (deviation from nominal confidence levels) and local power corrections.
Bartlett-type Corrections: By extending the expansion to order $n^{-1}$ , the authors identify the leading calibration error and construct Bartlett-type corrections to improve coverage accuracy.
Certified Computation: Recognizing that the projection often lacks a closed form, the paper develops a "certified localized dual algorithm." This algorithm reformulates the test decision as a distributionally robust optimization (DRO) problem, localizes the dual variables to a compact box where strong convexity/concavity holds, and provides rigorous bounds on numerical error to ensure the test decision is statistically valid.

3. Key Contributions and Results

A. Higher-Order Asymptotic Expansions

The paper derives a stochastic expansion for $n R_n(h)$ under the null and local alternatives:
$n R_n(h) = \langle V_n, \xi_n^{\otimes 2} \rangle + \frac{1}{\sqrt{n}} \langle K_n, \xi_n^{\otimes 3} \rangle + \varepsilon_n$
where $V_n$ and $K_n$ are empirical tensors derived from the Jacobian and Hessian of $h$ , and $\xi_n$ is a scaled empirical direction. The remainder $\varepsilon_n$ is uniformly bounded with high probability.

B. Size and Power Implications

Size Accuracy: The resulting Edgeworth theory shows that the plug-in WP test has a size error of order $\tilde{O}(n^{-1})$ . This is a significant improvement over the $O(n^{-1/2})$ error typical of first-order approximations.
Local Power: Under local alternatives, the paper identifies an explicit order- $n^{-1/2}$ correction to the local power.
Comparison with EL and $T^2$ : For scalar moments ( $d_h=1$ $d_{h} = 1$ ) under local location shifts, WP, EL, and Hotelling's $T^2$ $T^{2}$ share the same first-order local power. However, their second-order ranking differs:
- WP: The power correction is governed by a derivative-based curvature coefficient (involving the Hessian of $h$ ).
- EL: The power correction is governed by a value-based skewness coefficient (involving the third moment of $h$ ).
- $T^2$ : Has a zero second-order correction in this specific comparison regime.
  The paper demonstrates that WP can outperform EL and $T^2$ when the moment function exhibits favorable derivative curvature relative to its value skewness.

C. Bartlett-type Corrections

The authors extend the expansion to order $n^{-1}$ to construct two Bartlett-type corrections for scalar moment restrictions. These corrections remove the leading $O(n^{-1})$ calibration error, improving the coverage accuracy of confidence regions to $\tilde{O}(n^{-3/2})$ .

D. Certified Computation

The paper proposes Algorithm 1, a nested optimization scheme that:

Localizes the dual search space to a compact box where the inner problems are strongly convex.
Uses trisection and projected gradient ascent to solve the problem.
Provides a certified decision: the algorithm controls numerical error such that the probability of a "false rejection" or "false acceptance" due to computational inaccuracy is negligible.
Achieves a total computational complexity of $\tilde{O}(n)$ with high probability, matching the oracle test's statistical guarantees up to $\tilde{O}(n^{-1})$ .

4. Significance and Claims

The paper claims that its higher-order analysis is essential for understanding when the geometric nature of optimal transport (specifically the ground metric and local variation of the moment function) provides finite-sample advantages over reweighting-based methods like Empirical Likelihood.

Theoretical Insight: The work clarifies that the "transport geometry" of WP is not just a computational tool but a statistical feature that influences power through derivative-based curvature. This distinguishes WP from EL, which relies on value-based moments.
Practical Utility: The derived Bartlett corrections and the certified algorithm allow for more accurate finite-sample inference and reliable testing in settings where closed-form solutions are unavailable (e.g., nonlinear fairness constraints).
Application Context: The authors illustrate these findings using a fairness testing experiment. In this setting, moment restrictions encode equality of opportunity or odds. The experiment confirms the theoretical prediction that WP with an anisotropic transport metric can achieve higher power than EL or $T^2$ in regimes where the derivative curvature of the fairness constraint is favorable.

The paper concludes that while WP, EL, and $T^2$ are asymptotically equivalent at the first order, higher-order analysis reveals distinct finite-sample behaviors driven by the interplay between the moment function's geometry and the data distribution. The proposed certified algorithm ensures these theoretical gains are accessible in practice without compromising statistical validity.