Missing exposure information is a very common feature of many observational studies. Here we study identifiability and efficient estimation of causal effects on vector outcomes, in such cases where treatment is unconfounded but partially missing. We consider a missing at random setting where missingness in treatment can depend not only on complex covariates, but also on post-treatment outcomes. We give a new identifying expression for average treatment effects in this setting, along with the efficient influence function for this parameter in a nonparametric model, which yields a nonparametric efficiency bound. We use this latter result to construct nonparametric estimators that are less sensitive to the curse of dimensionality than usual, e. g. by having faster rates of convergence than the complex nuisance estimators they rely on. Further we show that these estimators can be root-n consistent and asymptotically normal under weak nonparametric conditions, even when constructed using flexible machine learning. Finally we apply these results to the problem of causal inference with a partially missing instrumental variable.

中文翻译：

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缺少曝光信息的有效非参数因果推断。

缺少接触信息是许多观察研究的一个非常普遍的特征。在这种情况下，我们研究了在治疗无混淆但部分缺失的情况下对向量结果的因果影响的可识别性和有效估计。我们认为随机设置中的缺失，其中治疗中的缺失不仅取决于复杂的协变量，而且还取决于治疗后的结果。我们给出了在这种情况下平均治疗效果的新识别表达式，以及在非参数模型中对此参数的有效影响函数，从而产生了非参数效率边界。我们使用后一个结果来构造非参数估计量，该估计量对维数诅咒的敏感性不如通常的e。G。比他们所依赖的复杂的干扰估计器具有更快的收敛速度。此外，我们证明了即使在使用灵活的机器学习方法构造的情况下，这些估计量在弱非参数条件下也可以是根n一致且渐近正态的。最后，我们将这些结果应用于因工具缺失而部分缺失的因果推理问题。