Deepening the Understanding of Double Robustness Geometrically

TL;DR

This paper explores the theoretical foundations of double robustness in estimators, revealing that convexity is key for influence curves to imply DR and providing a geometric interpretation using information geometry.

Contribution

It introduces a geometric perspective on double robustness, deriving conditions for the existence of DR estimators and highlighting the role of convexity.

Findings

Convexity enables influence curves to imply double robustness.

Necessary and sufficient conditions for DR estimator existence are derived.

A geometric interpretation of DR using information geometry concepts is provided.

Abstract

Double robustness (DR) is a widely-used property of estimators that provides protection against model misspecification and slow convergence of nuisance functions. Despite its widespread application, the theoretical foundation of DR remains underexplored. While DR is a property of global invariance along both nuisance directions, it is often implied by influence curves (ICs), which only have zero first-order derivatives in those directions locally. On the other hand, some literature proved the absence of DR estimating functions for the same estimand, under one parameterization yet was able to find one under another parameterization, highlighting the nuances in parameterization. In this short communication, we address two key questions: (1) Why do ICs frequently imply DR ``for free''? (2) Under what conditions would a given statistical model and parameterization support or prevent the existence of DR estimators? Using tools from semiparametric theory, we show that convexity is the crucial property that enables influence curves to imply DR. We then derive necessary and sufficient conditions for the existence of DR estimators. Our main contribution also lies in the novel geometric interpretation of DR using information geometry, a discipline devoted to integrating global differential geometry with statistical analysis. By leveraging concepts such as parallel transport, m-flatness, and m-curvature freeness, we characterize DR in terms of invariance along submanifolds. This geometric perspective deepens the understanding of when and why DR estimators exist.