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：我们提出了光滑希尔伯特值参数的估计方法，其中光滑性由路径可微性条件刻画。当参数空间为再生核希尔伯特空间时，我们提供了一种获得有效根n速率估计量及相应置信集的方法。这些估计量是基于希尔伯特值有效影响函数的交叉拟合一阶估计量的推广。即使使用任意干扰参数估计量（包括基于机器学习技术的估计量），我们也给出了理论保证。我们证明：只要参数具有有效影响函数，这些结果可自然推广到缺乏再生核的希尔伯特空间。然而，我们也揭示了一个令人遗憾的事实：当不存在再生核时，许多有趣的参数虽然路径可微，却缺乏有效影响函数。针对这些情形，我们提出了正则化一阶估计量及其关联置信集。我们还证明了路径可微性——本方法的核心要求——在多种情形下成立。具体而言，我们提供了多个路径可微参数的实例，并发展了相应的估计量与置信集。在这些实例中，有四个与因果推断学界当前研究尤为相关：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。