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.

翻译：双重稳健泛函的估计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和条件结果均值。我们研究如何通过估计干扰函数来获得双重稳健非参数泛函的最优收敛速率，该泛函在因果推断和条件独立性检验文献中具有广泛应用。针对若干插件型估计量和一步型估计量，我们阐明了干扰函数估计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。对于每种估计量和样本分割策略，我们证明了在低正则性条件下必须对干扰函数估计进行欠平滑处理，才能获得目标泛函的最优收敛速率。通过实施适当的干扰函数调优和样本分割策略，我们表明部分估计量能在干扰函数的所有H\"older光滑类中达到极小极大收敛速率。