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

Lin Liu, Rajarshi Mukherjee, James Robins

Abstract: 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 (ii) the class of functionals studied by Robins et al. (2008). The present state-of-the-art estimators for DR functionals ψ are double-machine-learning (DML) estimators (Chernozhukov et al., 2018). A DML estimator ψˆ1 of ψ depends on estimates pˆ(x) and bˆ(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−α) Wald confidence interval (CI) centered at ψˆ1. 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−α) 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 H0: "rate double-robustness holds", with non-trivial power against certain alternatives. If H0 is rejected, we will have falsified her justification. However, no assumption-lean test of H0, including ours, can be a consistent test. Thus, the failure of our test to reject is not meaningful evidence in favor of H0.