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.Comment: Preprint, under revie