  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.
