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：我们考虑在存在未观测混淆的情况下，使用近端因果推断工具（Miao等人[2018]，Tchetgen Tchetgen等人[2020]）对反事实的结果均值进行识别和推断。近端因果推断需要至少一个积分方程的解存在。我们通过展示假设积分方程的任意一个解存在，该解的一个线性函数（如其均值）的sqrt（n）-可估性需要另一个积分方程的解存在来激发对近端因果推断积分方程的存在性进行探讨。 积分方程的解可能不唯一，这使得估计和推断变得复杂。我们构造了一个一致的估计器，用于解决一个积分方程的解集，并且接着调整极值估计器理论来从估计的解集中找到一个一致的估计器，从而得到一个唯一定义的解。一个去偏估计量反事实均值被证明在额外的正则性条件下是根n一致的，正规的，并且在渐近半参数局部有效。