  Estimators of doubly robust functionals typically rely on estimating two
complex nuisance functions, such as the propensity score and conditional
outcome mean for the average treatment effect functional. We consider the
problem of how to estimate nuisance functions to obtain optimal rates of
convergence for a doubly robust nonparametric functional that has witnessed
applications across the causal inference and conditional independence testing
literature. For several plug-in type estimators and a one-step type estimator,
we illustrate the interplay between different tuning parameter choices for the
nuisance function estimators and sample splitting strategies on the optimal
rate of estimating the functional of interest. For each of these estimators and
each sample splitting strategy, we show the necessity to undersmooth the
nuisance function estimators under low regularity conditions to obtain optimal
rates of convergence for the functional of interest. By performing suitable
nuisance function tuning and sample splitting strategies, we show that some of
these estimators can achieve minimax rates of convergence in all H\"older
smoothness classes of the nuisance functions.
