Nuisance Function Tuning and Sample Splitting for Optimally Estimating a Doubly Robust Functional

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 estimators and a first-order bias-corrected 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 either undersmooth or oversmooth the nuisance function estimators under low regularity conditions to obtain optimal rates of convergence for the functional of interest. Unlike the existing literature, we show that plug-in and first-order bias-corrected estimators can achieve minimax rates of convergence across all H"older smoothness classes of the nuisance functions by careful combinations of sample splitting and nuisance function tuning strategies. We complement these results with numerical simulations illustrating the impact of different nuisance function tuning and sample splitting strategies.