We consider identification and inference about a counterfactual outcome mean when there is unmeasured confounding using tools from proximal causal inference (Miao et al. [2018], Tchetgen Tchetgen et al. [2020]). Proximal causal inference requires existence of solutions to at least one of two integral equations. We motivate the existence of solutions to the integral equations from proximal causal inference by demonstrating that, assuming the existence of a solution to one of the integral equations, $\sqrt{n}$-estimability of a linear functional (such as its mean) of that solution requires the existence of a solution to the other integral equation. Solutions to the integral equations may not be unique, which complicates estimation and inference. We construct a consistent estimator for the solution set for one of the integral equations and then adapt the theory of extremum estimators to find from the estimated set a consistent estimator for a uniquely defined solution. A debiased estimator for the counterfactual mean is shown to be root-$n$ consistent, regular, and asymptotically semiparametrically locally efficient under additional regularity conditions.

翻译：我们考虑在存在未测量混杂的情况下，利用近端因果推断工具（Miao et al. [2018]，Tchetgen Tchetgen et al. [2020]）对反事实结果均值进行识别与推断。近端因果推断要求至少存在两个积分方程中某一个的解。我们通过证明：假设其中一个积分方程的解存在，则该解的线性泛函（如其均值）的$\sqrt{n}$可估性需要另一个积分方程的解存在，从而为近端因果推断中积分方程解的存在性提供动机。积分方程的解可能不唯一，这给估计和推断带来了复杂性。我们为其中一个积分方程的解集构建了一致估计量，然后利用极值估计量理论从该估计集中找出唯一确定解的一致估计量。在附加正则条件下，反事实均值的去偏估计量被证明具有根号$n$一致性、正则性以及渐近半参数局部有效性。