We consider the problem of constructing minimax rate-optimal estimators for a doubly robust nonparametric functional that has witnessed applications across the causal inference and conditional independence testing literature. Minimax rate-optimal estimators for such functionals are typically constructed through higher-order bias corrections of plug-in and one-step type estimators and, in turn, depend on estimators of nuisance functions. In this paper, we consider a parallel question of interest regarding the optimality and/or sub-optimality of plug-in and one-step bias-corrected estimators for the specific doubly robust functional of interest. Specifically, we verify that by using undersmoothing and sample splitting techniques when constructing nuisance function estimators, one can achieve minimax rates of convergence in all H\"older smoothness classes of the nuisance functions (i.e. the propensity score and outcome regression) provided that the marginal density of the covariates is sufficiently regular. Additionally, by demonstrating suitable lower bounds on these classes of estimators, we demonstrate the necessity to undersmooth the nuisance function estimators to obtain minimax optimal rates of convergence.
翻译:我们考虑建造小型最大比率和最佳估计器的问题,以达到双重强健的非参数性功能,这种功能在因果推断和有条件独立测试文献中都得到了应用。这种功能的最小比率最佳估计器通常是通过对插头和一步型估计器进行更高层次的偏差校正来构建的,而反过来又取决于对扰动功能的偏差等级的估测器。在本文件中,我们考虑一个平行的利害问题,即插座和单步偏差修正估计器的最佳性和(或)次最佳性,但前提是我们变换器的边际密度是足够正常的。具体地说,我们核查,在建立偏差函数估计器时,通过使用下移动和样本分割技术,可以在所有H\"古老的扰动功能中达到最小趋同率的趋同率(即敏性评分和结果回归率),条件是我们变换器的边际密度是足够正常的。此外,我们通过展示这些测算器在最低等级下达到最优化的趋同功能,从而显示这些测算器的必要性。