The class of doubly-robust (DR) functionals studied by Rotnitzky et al. (2021) is of central importance in economics and biostatistics. It strictly includes both (i) the class of mean-square continuous functionals that can be written as an expectation of an affine functional of a conditional expectation studied by Chernozhukov et al. (2022b) and the class of functionals studied by Robins et al. (2008). The present state-of-the-art estimators for DR functionals $\psi$ are double-machine-learning (DML) estimators (Chernozhukov et al., 2018). A DML estimator $\widehat{\psi}_{1}$ of $\psi$ depends on estimates $\widehat{p} (x)$ and $\widehat{b} (x)$ of a pair of nuisance functions $p(x)$ and $b(x)$, and is said to satisfy "rate double-robustness" if the Cauchy--Schwarz upper bound of its bias is $o (n^{- 1/2})$. Were it achievable, our scientific goal would have been to construct valid, assumption-lean (i.e. no complexity-reducing assumptions on $b$ or $p$) tests of the validity of a nominal $(1 - \alpha)$ Wald confidence interval (CI) centered at $\widehat{\psi}_{1}$. But this would require a test of the bias to be $o (n^{-1/2})$, which can be shown not to exist. We therefore adopt the less ambitious goal of falsifying, when possible, an analyst's justification for her claim that the reported $(1 - \alpha)$ Wald CI is valid. In many instances, an analyst justifies her claim by imposing complexity-reducing assumptions on $b$ and $p$ to ensure "rate double-robustness". Here we exhibit valid, assumption-lean tests of $H_{0}$: "rate double-robustness holds", with non-trivial power against certain alternatives. If $H_{0}$ is rejected, we will have falsified her justification. However, no assumption-lean test of $H_{0}$, including ours, can be a consistent test. Thus, the failure of our test to reject is not meaningful evidence in favor of $H_{0}$.

翻译：Rotnitzky等人（2021）研究的双鲁棒（DR）泛函类在经济学和生物统计学中具有核心重要性。该泛函类严格包含：（i）Chernozhukov等人（2022b）研究的可写为条件期望的仿射泛函期望的均方连续泛函类，以及（ii）Robins等人（2008）研究的泛函类。当前DR泛函ψ的最先进估计量是双重机器学习（DML）估计量（Chernozhukov等人，2018）。ψ的DML估计量$\widehat{\psi}_{1}$依赖于一对干扰函数$p(x)$和$b(x)$的估计值$\widehat{p} (x)$和$\widehat{b} (x)$，若其偏差的柯西-施瓦茨上界为$o (n^{- 1/2})$，则称其满足“速率双鲁棒性”。若该性质可实现，我们的科学目标本应为构建有效的、假设精简（即不对b或p进行复杂性简化假设）的检验，用于验证以$\widehat{\psi}_{1}$为中心的标称$(1 - \alpha)$ Wald置信区间（CI）的有效性。但这需要检验偏差是否为$o (n^{-1/2})$，而可以证明此类检验不存在。因此，我们采取较为温和的目标：在可能情况下证伪分析师关于其报告的$(1 - \alpha)$ Wald CI有效的论证。在许多情况下，分析师通过施加关于b和p的复杂性简化假设以确保“速率双鲁棒性”来论证其主张。本文我们展示了有效的、假设精简的零假设$H_{0}$检验：“速率双鲁棒性成立”，并对某些备择假设具有非平凡检验功效。若拒绝$H_{0}$，则其论证被证伪。然而，包括我们的检验在内的任何假设精简的$H_{0}$检验都无法成为一致检验。因此，我们的检验未能拒绝$H_{0}$并不构成支持$H_{0}$的有意义证据。