W-Estimator Methods in Statistics

• W-Estimator refers to a collection of statistical estimators that utilize formulations based on Wasserstein distance, weighted likelihoods, and widely linear methods.
• The estimators achieve asymptotic normality and efficiency in well-specified models, with applications ranging from location-scale estimation to distributed and robust procedures.
• Implementation involves closed-form solutions in low dimensions and iterative numerical methods in higher dimensions, addressing challenges like covariate shift and computational scalability.

The term W-estimator encompasses several distinct statistical estimators termed 'W' in the literature, including those based on Wasserstein distance, statistical data depth, widely linear models, and related robust or distributed estimation paradigms. The specific definition depends on context, but in contemporary theoretical statistics, the most prominent meanings are: (i) Wasserstein (optimal transport) distance-based estimators for parametric models, (ii) minimum Wasserstein distance functionals under distributional shift, (iii) estimators exploiting weighted linear or likelihood structures, and (iv) widely linear unbiased estimators in linear models. This article synthesizes these main formulations, with precise formulations, asymptotic theory, and domain-specific applications.

1. Wasserstein Distance-based W-Estimators in Location-Scale Models

A canonical construction is the Wasserstein (W-) estimator for location-scale models, minimizing empirical 2-Wasserstein distance between the empirical distribution $\hat F_n$ and the model distribution $F(\cdot; \theta)$. For i.i.d. samples $x_1,\dots,x_n\in\mathbb R$ from the location-scale family

$$p(x; \theta) = \frac{1}{\sigma} f\left( \frac{x - \mu}{\sigma} \right), \quad \theta = (\mu, \sigma),$$

the estimator is defined as

$$\hat\theta_W = \operatorname{arg\,min}_\theta\int_0^1 [\hat F_n^{-1}(u) - F^{-1}(u; \theta)]^2\,du,$$

where $F^{-1}(\cdot; \theta)$ is the quantile function of the model (Amari, 2020, Amari et al., 2020).

Explicitly, letting $x_{(i)}$ denote the $i$th order statistic and $k_i$ be weights determined by the base density $f$,

$$\hat\mu_W = \frac{1}{n}\sum_{i=1}^n x_{(i)}, \qquad \hat\sigma_W = \sum_{i=1}^{n} k_i x_{(i)},$$

with $k_i = \int_{z_{i-1}}^{z_i} z f(z) dz$ and $z_i = F_0^{-1}(i/n)$, $F_0$ the CDF of $f$. $\hat\mu_W$ is the ordinary sample mean, while $\hat\sigma_W$ is an $L$-statistic with weights depending on $f$.

As $n\to\infty$, $(\hat\mu_W, \hat\sigma_W)$ are consistent for the true parameters, and $$(\sqrt{n}(\hat\mu_W - \mu_0), \sqrt{n}(\hat\sigma_W - \sigma_0))$$ is asymptotically normal; the explicit form of the asymptotic covariance matrix is provided in terms of moments of $f$ (Amari, 2020, Amari et al., 2020).

In the special case $f$ is standard Gaussian, $(\hat\mu_W, \hat\sigma_W)$ achieves the Cramér–Rao bound for $\mu$ and $\sigma$, i.e., it is Fisher efficient (Amari et al., 2020). For non-Gaussian $f$, the estimator for $\sigma$ has strictly larger asymptotic variance, indicating reduced (but stable) efficiency.

2. Wasserstein Score and Otto Information W-Estimators

Expanding from the univariate setting, the Wasserstein geometry provides an estimator via the Wasserstein score function—the vector field $\Phi(x; \theta)$ solving the continuity equation:

$$\frac{\partial}{\partial\theta_i} p(x; \theta) + \nabla_x \cdot [p(x; \theta) \nabla_x \Phi_i(x; \theta)] = 0, \qquad \mathbb{E}_\theta[\Phi_i(x; \theta)] = 0.$$

The (Wasserstein) W-estimator is defined as the root of the empirical score equations:

$$\sum_{t=1}^n \Phi_i(x_t; \hat\theta_W) = 0, \quad i = 1, \dots, p.$$

Under smoothness, identifiability, and regularity, $\hat\theta_W$ is asymptotically normal with covariance governed by the inverse Wasserstein (Otto) information matrix $G_W(\theta)$:

$$G_W(\theta)_{ij} = \mathbb{E}_\theta[\nabla_x \Phi_i(x; \theta) \cdot \nabla_x \Phi_j(x; \theta)].$$

When $G_W(\theta)$ is invertible, $$\sqrt{n}(\hat\theta_W - \theta) \xrightarrow{d} N(0, G_W(\theta)^{-1})$$, achieving the Wasserstein–Cramér–Rao lower bound (WCRLB) asymptotically. For location-scale families, the W-estimator is exactly the sample mean and sample standard deviation (Nishimori et al., 15 Jun 2025).

In scalar models, exact finite-sample W-efficiency is attained if and only if the family is an Otto e-geodesic, i.e., the score $\Phi(x; \theta)$ is constant in $\theta$ up to a reparametrization (Nishimori et al., 15 Jun 2025).

3. Minimum Wasserstein Distance Under Covariate Shift

Recent work formulates a minimum Wasserstein distance estimator (“W-estimator”) for population means under covariate shift, where only the marginal distribution of $X$ differs between labeled (source) and unlabeled (target) samples, but $Y|X$ is invariant. Given source data $(X_i, Y_i)$ and target covariates $\widetilde X_j$, the estimator finds weights

$$\hat p = \operatorname{arg\,min}_{p\in \Delta^n} W_2(F_1(\cdot; p), F_{1m}),$$

where $F_1(\cdot; p)$ is the candidate target marginal supported on source $X_i$ with weights $p_i$, and $F_{1m}$ is the empirical target marginal. The W-estimator for the target mean is then

$\hat\theta = \sum_{i=1}^n g(X_i, Y_i) \hat p_i,$

which, under suitable conditions, reduces to the standard 1-nearest neighbor estimator (Lang et al., 12 Jan 2026).

Notably, $\hat\theta$ is $\sqrt{m}$-consistent and asymptotically normal with limiting variance equal to $\operatorname{Var}_{F_1}(g(X,Y))$, but is not asymptotically linear (“irregular”), thus it may be super-efficient relative to the semiparametric efficient bound for regular estimators. Standard influence-function and bootstrap theory does not apply, and inference should use martingale CLT–based methods (Lang et al., 12 Jan 2026).

4. W-Estimator in Distributed and Robust Inference

Several distributed and robust statistical estimation procedures use the label "W-estimator" for weighted or widely-linear aggregates:

• First-Order Newton-type Estimator (FONE): In distributed convex loss minimization (possibly non-smooth), $w=\Sigma^{-1}v$ is the key object for one-step inference. The FONE directly approximates $\Sigma^{-1}v$ via stochastic iterative schemes, avoiding explicit Hessian computation and enabling valid inference for high-dimensional, non-smooth ERMs. The limiting distribution of $w^\top(\hat\theta - \theta^*)$ uses this estimator, with plug-in variance $\hat w^\top \hat A \hat w$, where $A$ is the score covariance (Chen et al., 2018).
• Weighted Distributed Estimator (WD-Estimator): In distributed M-estimation with heterogeneity, the WD estimator linearly aggregates local M-estimates with block-optimal weights,

$$\hat\phi^{\rm WD} = \left( \sum_k n_k H_k^{-1} \right)^{-1} \sum_k n_k H_k^{-1} \hat\phi_k,$$

where $H_k$ encodes the blockwise information and variance. The WD estimator achieves, and can improve upon, the statistical efficiency of the global M-estimator and GMM, while remaining communication-efficient. A bias-reduced (“debiased”) version extends applicability to much larger $K$ (number of machines) (Gu et al., 2022).

• Weighted Robust Estimator via Data Depth: The W-estimator in Agostinelli et al. (2024) solves depth-weighted likelihood equations,

$\sum_{i=1}^n w_i(\theta) s(X_i; \theta) = 0,$

where $w_i(\theta)$ depends on the statistical depth deviation between $X_i$ and the model. This estimator is consistent, asymptotically normal (with the same limiting variance as MLE in well-specified models), and achieves high robustness/breakdown point in elliptical families (Agostinelli et al., 22 Jul 2025).

5. Widely Linear Unbiased and $W^2$-Based Estimators in Signal Processing and Physics

In signal processing, widely linear unbiased estimators are designed for real-valued parameters embedded in complex measurement models. The best widely linear unbiased estimator (BWLUE) leverages both $y$ and $y^*$, yielding real-valued, unbiased outputs with strictly smaller variance than the classic BLUE (Lang et al., 2016):

$\hat x = E y + E^* y^*,$

with $E$ expressible in closed form. This estimator is optimal under proper Gaussian noise and full column rank $H$.

In neutrino physics, the $W^2$-estimator refers to a neutrino energy estimator based on the reconstructed final-state hadronic invariant mass. This estimator combines visible hadronic mass, lepton kinematics, and reconstructed proton counts to yield an energy estimate with small bias and robust performance across a range of interaction regimes (Thorpe et al., 14 Nov 2025).

6. Comparison and Domain-specific Performance

Variant	Core Principle	Context/Exemplar Papers
Wasserstein W	Minimize $W_2$ between empirical and model	(Amari, 2020, Amari et al., 2020)
Otto Score W	Wasserstein score, Otto information	(Nishimori et al., 15 Jun 2025)
Covariate W	OT-minimization under covariate shift	(Lang et al., 12 Jan 2026)
Weighted Dist.	Blockwise M-estimation weights	(Gu et al., 2022, Chen et al., 2018)
Depth-weighted	Depth-based likelihood weighting	(Agostinelli et al., 22 Jul 2025)
Widely Linear	Real-valued output, complex model	(Lang et al., 2016)
$W^2$-estimator	Hadronic invariant mass method	(Thorpe et al., 14 Nov 2025)

The theoretical and practical properties of these estimators depend on the statistical model, regularity, and regime of application. Wasserstein-based estimators are generally robust and often attain optimality in well-specified geometric settings, but may forgo Fisher efficiency outside these regimes. Weighted distributed and robust variants target computational and contamination resilience, prioritizing breakdown point and communication efficiency.

7. Implementation Considerations and Inference

For each variant, implementation is determined by the computational structure:

• Wasserstein distance-based estimators in 1D admit $O(n)$ closed-form; in higher dimensions, numerical or OT-solvers are required (Amari, 2020).
• Covariate shift W-estimators are efficiently reducible to $1$-NN search and simple averaging; in high dimensions, fast nearest-neighbor algorithms or approximations are crucial (Lang et al., 12 Jan 2026).
• Robust or depth-based W-estimators require depth computation per iteration; practical algorithms scale for $p=O(10)$, but approximate depths are needed for large $p$ (Agostinelli et al., 22 Jul 2025).
• Distributed W-estimators rely on local estimation and summary communication; structure preserves first-order efficiency with minimal communication (Gu et al., 2022).
• For FONE, step size, batch size, and number of rounds are tunable to balance statistical error, computational complexity, and communication (Chen et al., 2018).

Irregular estimators (as in covariate shift) demand nonstandard inference: plug-in Wald intervals with martingale CLT–based variance estimators are preferred over bootstrap or influence-function-based intervals, which may not be valid in non-asymptotically linear settings (Lang et al., 12 Jan 2026).

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References

(Chen et al., 2018, Amari, 2020, Amari et al., 2020, Gu et al., 2022, Nishimori et al., 15 Jun 2025, Agostinelli et al., 22 Jul 2025, Thorpe et al., 14 Nov 2025, Lang et al., 12 Jan 2026, Lang et al., 2016)