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 Hölder 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.

翻译：双重稳健泛函的估计器通常依赖于估计两个复杂的干扰函数，例如平均处理效应泛函中的倾向得分与条件结果均值。本文研究如何通过干扰函数估计来获得最优收敛速率，该问题针对已在因果推断与条件独立性检验文献中得到广泛应用的双重稳健非参数泛函。针对若干插件估计器及一阶偏差校正估计器，我们阐明了干扰函数估计器中不同调参选择与样本分割策略对目标泛函最优估计速率的交互影响。对于每种估计器及样本分割策略，我们证明了在低正则性条件下，为获得目标泛函的最优收敛速率，必须对干扰函数估计器实施欠平滑或过平滑处理。与现有文献不同，本文通过样本分割与干扰函数调参策略的精细组合，证明了插件估计器与一阶偏差校正估计器能够在干扰函数的所有Hölder光滑度类别中达到极小极大收敛速率。我们通过数值模拟补充了这些结果，展示了不同干扰函数调参与样本分割策略的实际影响。