Minimum Wasserstein distance estimator under covariate shift: closed-form, super-efficiency and irregularity

TL;DR

This paper introduces a novel Wasserstein distance-based estimator for covariate shift problems that is simple, closed-form, and exhibits super-efficiency, outperforming traditional methods in simulations and real data.

Contribution

It proposes a new W-estimator for covariate shift that avoids explicit modeling, has a closed-form expression, and demonstrates super-efficiency and irregularity properties.

Findings

The W-estimator is numerically equivalent to 1-nearest neighbor.

It achieves root-n asymptotic normality but is not asymptotically linear.

Numerical simulations and real data analysis show its superior performance.

Abstract

Covariate shift arises when covariate distributions differ between source and target populations while the conditional distribution of the response remains invariant, and it underlies problems in missing data and causal inference. We propose a minimum Wasserstein distance estimation framework for inference under covariate shift that avoids explicit modeling of outcome regressions or importance weights. The resulting W-estimator admits a closed-form expression and is numerically equivalent to the classical 1-nearest neighbor estimator, yielding a new optimal transport interpretation of nearest neighbor methods. We establish root-$n$ asymptotic normality and show that the estimator is not asymptotically linear, leading to super-efficiency relative to the semiparametric efficient estimator under covariate shift in certain regimes, and uniformly in missing data problems. Numerical simulations, along with an analysis of a rainfall dataset, underscore the exceptional performance of our W-estimator.