We study the problem of constructing an estimator of the average treatment effect (ATE) that exhibits doubly-robust asymptotic linearity (DRAL). This is a stronger requirement than doubly-robust consistency. A DRAL estimator can yield asymptotically valid Wald-type confidence intervals even when the propensity score or the outcome model is inconsistently estimated. On the contrary, the celebrated doubly-robust, augmented-IPW (AIPW) estimator generally requires consistent estimation of both nuisance functions for standard root-n inference. We make three main contributions. First, we propose a new hybrid class of distributions that consists of the structure-agnostic class introduced in Balakrishnan et al (2023) with additional smoothness constraints. While DRAL is generally not possible in the pure structure-agnostic class, we show that it can be attained in the new hybrid one. Second, we calculate minimax lower bounds for estimating the ATE in the new class, as well as in the pure structure-agnostic one. Third, building upon the literature on doubly-robust inference (van der Laan, 2014, Benkeser et al, 2017, Dukes et al 2021), we propose a new estimator of the ATE that enjoys DRAL. Under certain conditions, we show that its rate of convergence in the new class can be much faster than that achieved by the AIPW estimator and, in particular, matches the minimax lower bound rate, thereby establishing its optimality. Finally, we clarify the connection between DRAL estimators and those based on higher-order influence functions (Robins et al, 2017) and complement our theoretical findings with simulations.

翻译：我们研究构建具有双鲁棒渐近线性性(DRAL)的平均处理效应(ATE)估计量问题。这是一个比双鲁棒相合性更强的要求。即使倾向得分或结果模型被不一致地估计，DRAL估计量仍能产生渐近有效的Wald型置信区间。相反，著名的双鲁棒增强型逆概率加权(AIPW)估计量通常需要对两个干扰函数的一致估计才能进行标准的根号n推断。我们做出三项主要贡献。首先，我们提出一个新的混合分布类，它由Balakrishnan等人(2023)引入的结构无感知类加上额外光滑性约束构成。虽然在纯结构无感知类中通常无法实现DRAL，但我们证明它可以在新的混合类中实现。其次，我们计算了在新类及纯结构无感知类中估计ATE的极小化最大下界。第三，基于双鲁棒推断的文献(van der Laan, 2014; Benkeser等人, 2017; Dukes等人, 2021)，我们提出一个具有DRAL性质的新ATE估计量。在一定条件下，我们证明该估计量在新类中的收敛速度可远快于AIPW估计量，且特别地达到了极小化最大下界速率，从而证明了其最优性。最后，我们阐明了DRAL估计量与基于高阶影响函数的估计量(Robins等人, 2017)之间的联系，并通过模拟补充了理论结果。