Wasserstein-Cramér-Rao Theory of Unbiased Estimation (2511.07414v1)

Abstract: The quantity of interest in the classical Cramér-Rao theory of unbiased estimation (e.g., the Cramér-Rao lower bound, its exact attainment for exponential families, and asymptotic efficiency of maximum likelihood estimation) is the variance, which represents the instability of an estimator when its value is compared to the value for an independently-sampled data set from the same distribution. In this paper we are interested in a quantity which represents the instability of an estimator when its value is compared to the value for an infinitesimal additive perturbation of the original data set; we refer to this as the "sensitivity" of an estimator. The resulting theory of sensitivity is based on the Wasserstein geometry in the same way that the classical theory of variance is based on the Fisher-Rao (equivalently, Hellinger) geometry, and this insight allows us to determine a collection of results which are analogous to the classical case: a Wasserstein-Cramér-Rao lower bound for the sensitivity of any unbiased estimator, a characterization of models in which there exist unbiased estimators achieving the lower bound exactly, and some concrete results that show that the Wasserstein projection estimator achieves the lower bound asymptotically. We use these results to treat many statistical examples, sometimes revealing new optimality properties for existing estimators and other times revealing entirely new estimators.

• The paper introduces a new sensitivity measure for unbiased estimators using Wasserstein metrics to capture local perturbation effects.
• It establishes lower bounds on sensitivity analogous to the classical Cramér-Rao variance bound, with concrete examples across models.
• The work reveals that variance-optimal estimators can differ from sensitivity-efficient ones, guiding robust estimator design.

Wasserstein-Cramér-Rao Theory of Unbiased Estimation: Geometric Foundations and Sensitivity-Efficiency

Introduction and Motivation

The classical Cramér-Rao framework measures the instability of unbiased estimators through their variance, reflecting how estimates fluctuate under independent resampling. The present work extends this viewpoint by introducing a new instability measure: sensitivity, capturing the change in an estimator's value under infinitesimal perturbations to the data. This notion is motivated by practical situations involving measurement errors or deliberate perturbations (e.g., for robustness or privacy).

The key insight of the paper is to frame sensitivity in terms of Wasserstein geometry, paralleling the classical relationship between variance and Fisher-Rao (or Hellinger) geometry. Specifically, the authors develop a Wasserstein-Cramér-Rao theory that provides lower bounds on sensitivity for unbiased estimators, characterizes attainability of these bounds, and reveals the statistical optimality of certain estimators in this new geometric framework.

Sensitivity: Definition and Conceptual Role

Formally, given a parametric family $$\{P_\theta: \theta \in \Theta\} \subseteq \mathcal{P}_2(\mathbb{R}^d)$$ and an estimator $T_n: (\mathbb{R}^d)^n \to \mathbb{R}$, the sensitivity is defined as

$$\operatorname{Sen}_P(T_n) = \mathbb{E}_{P}\left[\sum_{i=1}^n \|\nabla_{x_i} T_n(X_1, ..., X_n)\|^2\right].$$

This quantifies the estimator’s Dirichlet energy, or its first-order reaction to infinitesimal additive noise, and can be seen as a limiting case of the perturbative sensitivity to small, independent Gaussian measurement errors.

This is fundamentally different from variance: the latter assesses stability with respect to complete resampling, while sensitivity quantifies local stability to data perturbation. Notably, certain estimators are variance-efficient but have poor sensitivity—an effect that is apparent, for example, in non-differentiable estimators like order statistics.

Illustrative Example: Uniform Scale Family

To visualize this contrast, the paper investigates the uniform scale model, where $P_\theta = \operatorname{Unif}[0, \theta]$. Consider these three estimators of $\theta$:

• MLE: $T_n^\text{MLE} = \max_i X_i$
• Best linear estimator (BLE): $T_n^\text{BLE} = 2 n^{-1}\sum X_i$
• Wasserstein projection estimator (WPE): $T_n^\text{WPE}$

The variance and sensitivity behave markedly differently across these estimators:

Figure 1: Sensitivity and variance for three estimators in the uniform scale family. MLE achieves variance of $O(n^{-2})$ but constant sensitivity; BLE and WPE both exhibit $O(n^{-1})$ scaling, but WPE yields a smaller prefactor for both metrics.

Geometric Foundations: Fisher-Rao vs. Wasserstein

The classical Cramér-Rao lower bound is intrinsically linked to the Fisher-Rao geometry on the space of probability measures, where the Fisher information quantifies infinitesimal changes of the likelihood. In contrast, the new sensitivity theory is grounded in the Wasserstein metric, relying on optimal transport to describe how probability masses shift under parameter perturbation.

For parametric models, the analog of the Fisher score function, $G_\theta$, is the transport linearization $\Phi_\theta$, defined as the derivative of the optimal transport map between $P_\theta$ and a nearby $P_{\theta+th}$. The Wasserstein information matrix $J(\theta)$ then becomes

$$J(\theta) = \mathbb{E}_{P_\theta}\left[\Phi_\theta(X)^\top \Phi_\theta(X)\right].$$

A parametric model is said to be differentiable in the Wasserstein sense (DWS) if the optimal transport maps vary smoothly in $\theta$.

This geometric structure enables the derivation of Wasserstein-Cramér-Rao lower bounds on the sensitivity of unbiased estimators, efficiently paralleling the classical variance theory.

Wasserstein-Cramér-Rao Lower Bound

Under regularity (the DWS condition and technical smoothness/integrability), the paper establishes the central result: $$\operatorname{Sen}_\theta(T_n) \geq \frac{(\chi'(\theta))^2}{n J(\theta)},$$ for any unbiased estimator $T_n$ of $\chi(\theta)$, mirroring the Cramér-Rao variance bound but replacing the Fisher information with the Wasserstein information.

A detailed analogy is established:

• Variance $\leftrightarrow$ Fisher-Rao geometry, Cramér-Rao bound
• Sensitivity $\leftrightarrow$ Wasserstein geometry, Wasserstein-Cramér-Rao bound
• Exponential families $\leftrightarrow$ transport families

Attainability and Characterization of Sensitivity-Efficient Estimators

The paper develops necessary and sufficient conditions for equality in the sensitivity bound, introducing the notion of transport families. Analogous to exponential families (where the Fisher-Rao bound is tight), transport families admit estimators achieving the Wasserstein-Cramér-Rao bound exactly.

In these families, estimators of the form

$$T_n(X_1,...,X_n) = \frac{1}{n}\sum_{i=1}^n \phi(X_i)$$

for suitable potentials $\phi$ are unbiased and sensitivity-efficient for specific parameterizations of interest. The potential function is, in a sense, the Wasserstein-theoretic counterpart to sufficient statistics.

Contrast is made with the classical setting: some variance-efficient estimators are not sensitivity-efficient, and vice versa—a novel and non-trivial distinction.

Asymptotic Sensitivity-Efficiency via Wasserstein Projection

For parametric families that are not transport families, the paper investigates the Wasserstein projection estimator (WPE), defined as the best-fit parameter in Wasserstein distance: $$T_n^\text{WPE} = \arg\min_{\theta \in \Theta} W_2^2\left(P_\theta, \bar{P}_n\right),$$ where $\bar{P}_n$ is the empirical measure.

The authors prove that, under suitable regularity and identifiability conditions, the WPE is asymptotically sensitivity-efficient—its sensitivity approaches the theoretical lower bound at rate $O(1/n)$. This result is exact in dimension one (with explicit characterizations using quantile calculus) and extends, under technical assumptions, to higher dimensions (with additional geometric regularity required for the empirical power/transport cells).

These findings generalize the role of the MLE in Fisher-Rao information geometry: while the MLE minimizes asymptotic variance, the WPE minimizes asymptotic sensitivity under Wasserstein geometry.

Concrete Statistical Examples

Several canonical models are analyzed.

• Gaussian Location Family: The sample mean is both variance- and sensitivity-efficient.
• Uniform Scale Family: The BLE is asymptotically optimal for both criteria, but the WPE dominates in terms of constants—demonstrated numerically and theoretically.
• Laplace Location Family: The sample median is variance-efficient but not sensitivity-efficient; the sample mean, though suboptimal in variance, has optimal sensitivity.
• Pareto and Scale Families: The squared norm statistic, not the parameter $\theta$ itself, is the estimand for which sensitivity-efficiency is achievable.

These examples highlight that sensitivity-optimal estimators may differ substantially from classical optimal estimators in both their algebraic form and their statistical properties.

Implications, Theory, and Future Directions

Theoretical Significance

The paper demonstrates a deep geometric duality between variance and sensitivity, linking optimal estimation to the differential geometry of the parameter-induced submanifold under two distinct metrics. It establishes that minimizing local instability to data perturbations (sensitivity) yields fundamentally different optimal estimators compared to resampling-based instability (variance).

The authors provide tools for constructing and analyzing estimators that are robust to measurement error and other perturbative effects. In doing so, they clarify the role of Wasserstein geometry in statistical estimation theory, which is significant given the increasing relevance of optimal transport in statistics, machine learning, and applied mathematics.

Practical Implications

Sensitivity-efficient estimators are natural candidates for applications where data collection involves inevitable, small-scale noise—or when deliberate perturbations (e.g., differential privacy mechanisms, randomized smoothing for stability, simulation-based calibration) are employed. The theory directly quantifies the minimum performance degradation resulting from such perturbations and offers constructive recipes for estimator design.

Perspectives for AI and Statistics

This work opens new questions regarding optimal statistical procedures in regimes where stability to local noise is the primary desideratum rather than overall sampling variability. There are ramifications for robust statistics, privacy-preserving learning algorithms, and distributionally robust optimization, as Wasserstein balls are now widely used in uncertainty quantification and model misspecification.

A rich direction for future work is the extension of these results to:

• Discrete spaces and models where optimal transport is combinatorial and the smooth structure breaks down.
• Nonparametric settings and models with singularities or heterogenous supports.
• Understanding the behavior and computation of WPE in high-dimensional settings when strong regularity cannot be guaranteed.

Lastly, the link between geometric stability and the predictability-computability-stability (PCS) framework hints at broader foundational principles unifying statistical theory and practice.

Conclusion

The Wasserstein-Cramér-Rao theory of this paper provides an internally coherent, geometrically grounded framework for understanding and optimizing the sensitivity of unbiased estimators. Through analogies and contrasts with the classical variance-oriented theory, it elucidates new types of statistical optimality that are directly relevant to contemporary problems with noise, privacy, and robustness constraints. This work will inform both the practice and theory of statistical inference in settings where perturbative stability is paramount.