Proximal causal learning is a promising framework for identifying the causal effect under the existence of unmeasured confounders. Within this framework, the doubly robust (DR) estimator was derived and has shown its effectiveness in estimation, especially when the model assumption is violated. However, the current form of the DR estimator is restricted to binary treatments, while the treatment can be continuous in many real-world applications. The primary obstacle to continuous treatments resides in the delta function present in the original DR estimator, making it infeasible in causal effect estimation and introducing a heavy computational burden in nuisance function estimation. To address these challenges, we propose a kernel-based DR estimator that can well handle continuous treatments. Equipped with its smoothness, we show that its oracle form is a consistent approximation of the influence function. Further, we propose a new approach to efficiently solve the nuisance functions. We then provide a comprehensive convergence analysis in terms of the mean square error. We demonstrate the utility of our estimator on synthetic datasets and real-world applications.

翻译：近端因果学习是在存在未测量混杂因素的情况下识别因果效应的一个具有前景的框架。在该框架下，双重稳健（DR）估计量被推导出来，并在模型假设被违反时仍能展现其估计有效性。然而，目前DR估计量的形式仅适用于二元治疗，而许多实际应用中的治疗变量可以是连续的。连续治疗面临的主要障碍在于原始DR估计量中存在的delta函数，这使得它在因果效应估计中不可行，并在 nuisance 函数估计中引入了沉重的计算负担。为应对这些挑战，我们提出了一种基于核的DR估计量，该估计量能够有效处理连续治疗。凭借其平滑性，我们证明其oracle形式是影响函数的一致近似。此外，我们提出了一种高效求解nuisance函数的新方法。随后，我们给出了关于均方误差的综合收敛性分析。我们在合成数据集和实际应用中展示了该估计量的实用性。