Assumption-lean falsification tests of rate double-robustness of double-machine-learning estimators

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 $H_{0}$.