Double robustness (DR) is a widely-used property of estimators that provides protection against model misspecification and slow convergence of nuisance functions. While DR is a global property on the probability distribution manifold, it often coincides with influence curves, which only ensure orthogonality to nuisance directions locally. This apparent discrepancy raises fundamental questions about the theoretical underpinnings of DR. In this short communication, we address two key questions: (1) Why do influence curves frequently imply DR "for free"? (2) Under what conditions do DR estimators exist for a given statistical model and parameterization? 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 under a mean squared differentiable path-connected parameterization. Our main contribution also lies in the novel geometric interpretation of DR using information geometry. 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. The results not only resolve apparent mysteries surrounding DR but also have practical implications for the construction and analysis of DR estimators. The geometric insights open up new connections and directions for future research. Our findings aim to solidify the theoretical foundations of a fundamental concept and contribute to the broader understanding of robust estimation in statistics.

翻译：双重稳健性（Double Robustness, DR）是估计量广泛使用的性质，可防范模型误设及干扰函数的缓慢收敛。尽管DR是概率分布流形上的全局性质，但它常与仅局部确保干扰方向正交性的影响曲线重合。这一表面矛盾引发了对DR理论基础的根本性质疑。本短讯中，我们探讨两个关键问题：（1）为何影响曲线常"自动"蕴含DR？（2）在给定统计模型与参数化下，DR估计量存在的条件是什么？借助半参数理论工具，我们证明凸性是使影响曲线蕴含DR的关键性质，继而推导出在均方可微路径连通参数化下DR估计量存在的充要条件。主要贡献还在于利用信息几何对DR进行新颖的几何解读。通过运用平行传输、m-平坦性及m-曲率自由性等概念，我们以子流形上的不变性刻画DR。这一几何视角深化了对DR估计量存在时机与原因的理解。研究结果不仅解开了围绕DR的表层谜团，还对DR估计量的构造与分析具有实际意义。这些几何洞见为未来研究开辟了新联系与新方向。我们的发现旨在夯实这一基本概念的理论基础，并促进对统计学中稳健估计的更广泛理解。