  A moment function is called doubly robust if it is comprised of two nuisance
functions and the estimator based on it is a consistent estimator of the target
parameter even if one of the nuisance functions is misspecified. In this paper,
we consider a class of doubly robust moment functions originally introduced in
(Robins et al., 2008). We demonstrate that this moment function can be used to
construct estimating equations for the nuisance functions. The main idea is to
choose each nuisance function such that it minimizes the dependency of the
expected value of the moment function to the other nuisance function. We
implement this idea as a minimax optimization problem. We then provide
conditions required for asymptotic linearity of the estimator of the parameter
of interest, which are based on the convergence rate of the product of the
errors of the nuisance functions, as well as the local ill-posedness of a
conditional expectation operator. The convergence rates of the nuisance
functions are analyzed using the modern techniques in statistical learning
theory based on the Rademacher complexity of the function spaces. We
specifically focus on the case that the function spaces are reproducing kernel
Hilbert spaces, which enables us to use its spectral properties to analyze the
convergence rates. As an application of the proposed methodology, we consider
the parameter of average causal effect both in presence and absence of latent
confounders. For the case of presence of latent confounders, we use the
recently proposed proximal causal inference framework of (Miao et al., 2018;
Tchetgen Tchetgen et al., 2020), and hence our results lead to a robust
non-parametric estimator for average causal effect in this framework.
