In a variety of applications, including nonparametric instrumental variable (NPIV) analysis, proximal causal inference under unmeasured confounding, and missing-not-at-random data with shadow variables, we are interested in inference on a continuous linear functional (e.g., average causal effects) of nuisance function (e.g., NPIV regression) defined by conditional moment restrictions. These nuisance functions are generally weakly identified, in that the conditional moment restrictions can be severely ill-posed as well as admit multiple solutions. This is sometimes resolved by imposing strong conditions that imply the function can be estimated at rates that make inference on the functional possible. In this paper, we study a novel condition for the functional to be strongly identified even when the nuisance function is not; that is, the functional is amenable to asymptotically-normal estimation at $\sqrt{n}$-rates. The condition implies the existence of debiasing nuisance functions, and we propose penalized minimax estimators for both the primary and debiasing nuisance functions. The proposed nuisance estimators can accommodate flexible function classes, and importantly they can converge to fixed limits determined by the penalization regardless of the identifiability of the nuisances. We use the penalized nuisance estimators to form a debiased estimator for the functional of interest and prove its asymptotic normality under generic high-level conditions, which provide for asymptotically valid confidence intervals. We also illustrate our method in a novel partially linear proximal causal inference problem and a partially linear instrumental variable regression problem.

翻译：在非参数工具变量（NPIV）分析、存在未测量混杂的近端因果推断，以及具有影子变量的非随机缺失数据等各类应用中，我们关注由条件矩约束定义的干扰函数（如NPIV回归）的连续线性泛函（如平均因果效应）的推断问题。这些干扰函数通常被弱识别，表现为条件矩约束可能具有严重不适定性并允许多个解。有时通过施加强条件来避免这一问题，即要求干扰函数能以支持泛函推断的速率被估计。本文研究一种新颖条件，使得即便干扰函数未被强识别，其泛函仍能被强识别——即该泛函能以$\sqrt{n}$速率实现渐近正态估计。该条件表明存在去偏干扰函数，我们为原始干扰函数和去偏干扰函数均提出惩罚极小极大估计量。所提干扰估计量可适应灵活的函数类，且关键在于无论干扰函数的可识别性如何，它们都能收敛至由惩罚项决定的固定极限。我们利用惩罚干扰估计量构建目标泛函的去偏估计量，并在通用高阶条件下证明其渐近正态性，从而提供渐近有效的置信区间。此外，我们还在新颖的偏线性近端因果推断问题和偏线性工具变量回归问题中展示了该方法的应用。