We present estimators for smooth Hilbert-valued parameters, where smoothness is characterized by a pathwise differentiability condition. When the parameter space is a reproducing kernel Hilbert space, we provide a means to obtain efficient, root-n rate estimators and corresponding confidence sets. These estimators correspond to generalizations of cross-fitted one-step estimators based on Hilbert-valued efficient influence functions. We give theoretical guarantees even when arbitrary estimators of nuisance functions are used, including those based on machine learning techniques. We show that these results naturally extend to Hilbert spaces that lack a reproducing kernel, as long as the parameter has an efficient influence function. However, we also uncover the unfortunate fact that, when there is no reproducing kernel, many interesting parameters fail to have an efficient influence function, even though they are pathwise differentiable. To handle these cases, we propose a regularized one-step estimator and associated confidence sets. We also show that pathwise differentiability, which is a central requirement of our approach, holds in many cases. Specifically, we provide multiple examples of pathwise differentiable parameters and develop corresponding estimators and confidence sets. Among these examples, four are particularly relevant to ongoing research by the causal inference community: the counterfactual density function, dose-response function, conditional average treatment effect function, and counterfactual kernel mean embedding.

翻译：本文针对光滑希尔伯特值参数提出了估计量，其光滑性由路径可微条件刻画。当参数空间为再生核希尔伯特空间时，我们提供了获取有效、根n速率估计量及相应置信集的方法。这些估计量对应于基于希尔伯特值有效影响函数的交叉拟合单步估计量的推广。即使使用任意干扰函数估计量（包括基于机器学习技术的估计量），我们仍给出了理论保证。研究结果表明，只要参数具有有效影响函数，这些结果可自然延伸至缺乏再生核的希尔伯特空间。然而我们也发现一个不利事实：当不存在再生核时，许多有趣的参数虽然满足路径可微性，却无法具备有效影响函数。针对此类情形，我们提出正则化单步估计量及其置信集。此外，我们证明了作为方法核心要求的路径可微性在众多场景中成立：具体提供了多个路径可微参数示例，并开发了相应估计量与置信集。在这些示例中，四个与因果推断领域的持续研究高度相关：反事实密度函数、剂量反应函数、条件平均处理效应函数及反事实核均值嵌入。