  Robins et al. (2008) introduced a class of influence functions (IFs) which
could be used to obtain doubly robust moment functions for the corresponding
parameters. However, that class does not include the IF of parameters for which
the nuisance functions are solutions to integral equations. Such parameters are
particularly important in the field of causal inference, specifically in the
recently proposed proximal causal inference framework of Tchetgen Tchetgen et
al. (2020), which allows for estimating the causal effect in the presence of
latent confounders. In this paper, we first extend the class of Robins et al.
to include doubly robust IFs in which the nuisance functions are solutions to
integral equations. Then we demonstrate that the double robustness property of
these IFs can be leveraged to construct estimating equations for the nuisance
functions, which enables us to solve the integral equations without resorting
to parametric models. We frame the estimation of the nuisance functions as a
minimax optimization problem. We provide convergence rates for the nuisance
functions and conditions required for asymptotic linearity of the estimator of
the parameter of interest. The experiment results demonstrate that our proposed
methodology leads to robust and high-performance estimators for average causal
effect in the proximal causal inference framework.
