Liu Lin, Mukherjee Rajarshi, Robins James M
Institute of Natural Sciences, MOE-LSC, School of Mathematical Sciences, CMA-Shanghai, SJTU-Yale Joint Center for Biostatistics and Data Science, Shanghai Jiao Tong University; Shanghai Artificial Intelligence Laboratory.
Department of Biostatistics, Harvard University.
J Econom. 2024 Mar;240(2). doi: 10.1016/j.jeconom.2023.105500. Epub 2023 Aug 19.
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 are double-machine-learning (DML) estimators (Chernozhukov et al., 2018). A DML estimator of depends on estimates and of a pair of nuisance functions and , and is said to satisfy "rate double-robustness" if the Cauchy-Schwarz upper bound of its bias is . Were it achievable, our scientific goal would have been to construct valid, assumption-lean (i.e. no complexity-reducing assumptions on or ) tests of the validity of a nominal () Wald confidence interval (CI) centered at . But this would require a test of the bias to be , 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 () Wald CI is valid. In many instances, an analyst justifies her claim by imposing complexity-reducing assumptions on and to ensure "rate double-robustness". Here we exhibit valid, assumption-lean tests of : "rate double-robustness holds", with non-trivial power against certain alternatives. If is rejected, we will have falsified her justification. However, no assumption-lean test of , including ours, can be a consistent test. Thus, the failure of our test to reject is not meaningful evidence in favor of .
Rotnitzky等人（2021年）研究的双稳健（DR）泛函类在经济学和生物统计学中具有核心重要性。它严格包含以下两类：（i）可写成Chernozhukov等人（2022b年）研究的条件期望的仿射泛函期望的均方连续泛函类，以及Robins等人（2008年）研究的泛函类。目前DR泛函的最优估计量是双机器学习（DML）估计量（Chernozhukov等人，2018年）。DR泛函的DML估计量取决于一对干扰函数的估计量和，并且如果其偏差的柯西 - 施瓦茨上界为，则称其满足“速率双稳健性”。如果能够实现，我们的科学目标本应是构建关于以 为中心的名义（）Wald置信区间（CI）有效性的有效、假设较少（即对或无复杂度降低假设）的检验。但这需要一个偏差为的检验，而这是不存在的。因此，我们采用了一个不那么雄心勃勃的目标，即尽可能证伪分析师声称所报告的（）Wald CI有效的理由。在许多情况下，分析师通过对和施加复杂度降低假设来确保“速率双稳健性”，以此来证明其声称的合理性。在这里，我们展示了关于“速率双稳健性成立”的有效、假设较少的检验，对某些备择假设具有非平凡的检验功效。如果被拒绝，我们就证伪了她的理由。然而，包括我们的检验在内，没有任何假设较少的检验可以是一致检验。因此，我们的检验未能拒绝并不是支持的有意义证据。
