Nonparametric causal inference with confounders missing not at random

We consider the estimation and inference of Average Causal Effects (ACE) when confounders are missing not at random. The identification has been discussed in literature; however, limited effort has been devoted into developing feasible nonparametric inference methods. The primary challenge arises from the estimation process of the missingness mechanism, an ill-posed problem that poses obstacles in establishing asymptotic theory. This paper contributes to filling this gap in the following ways. Firstly, we introduce a weak pseudo-metric to guarantee a faster convergence rate of the missingness mechanism estimator. Secondly, we employ a representer to derive the explicit expression of the influence function. We also propose a practical and stable approach to estimate the variance and construct the confidence interval. We verify our theoretical results in the simulation studies.