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 ${\hat{\psi }}_1$ of $\psi$ depends on estimates $\hat{p}\left(x\right)$ and $\hat{b}\left(x\right)$ of a pair of nuisance functions $p\left(x\right)$ and $b\left(x\right)$, and is said to satisfy "rate double-robustness" if the Cauchy-Schwarz upper bound of its bias is $o\left(n^{-1/2}\right)$. Were it achievable, our scientific goal would have been to construct valid, assumption-lean (i.e. no complexity-reducing assumptions on $b$ oder $p$) tests of the validity of a nominal ($1-\alpha$) Wald confidence interval (CI) centered at ${\hat{\psi }}_1$. But this would require a test of the bias to be $o\left(n^{-1/2}\right)$, 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 ${\text{H}}_{\text{0}}$: "rate double-robustness holds", with non-trivial power against certain alternatives. If ${\text{H}}_{\text{0}}$ is rejected, we will have falsified her justification. However, no assumption-lean test of ${\text{H}}_{\text{0}}$, including ours, can be a consistent test. Thus, the failure of our test to reject is not meaningful evidence in favor of ${\text{H}}_{\text{0}}$.