We present and study semi-parametric estimators for the mean of functional outcomes in situations where some of these outcomes are missing and covariate information is available on all units. Assuming that the missingness mechanism depends only on the covariates (missing at random assumption), we present two estimators for the functional mean parameter, using working models for the functional outcome given the covariates, and the probability of missingness given the covariates. We contribute by establishing that both these estimators have Gaussian processes as limiting distributions and explicitly give their covariance functions. One of the estimators is double robust in the sense that the limiting distribution holds whenever at least one of the nuisance models is correctly specified. These results allow us to present simultaneous confidence bands for the mean function with asymptotically guaranteed coverage. A Monte Carlo study shows the finite sample properties of the proposed functional estimators and their associated simultaneous inference. The use of the method is illustrated in an application where the mean of counterfactual outcomes is targeted.

翻译：本文提出并研究了在部分功能型结果缺失且所有单元均具有协变量信息的情况下，对功能型结果均值进行半参数估计的方法。假设缺失机制仅依赖于协变量（随机缺失假设），我们利用功能型结果给定协变量的工作模型以及缺失概率给定协变量的工作模型，提出了两种针对功能型均值参数的估计量。我们的主要贡献在于证明了这两种估计量均以高斯过程作为极限分布，并明确给出了其协方差函数。其中一种估计量具有双稳健性，即只要至少一个干扰模型被正确设定，其极限分布就成立。这些结果使我们能够构建具有渐近保证覆盖率的均值函数同步置信带。蒙特卡洛研究展示了所提出的功能型估计量及其同步推断方法的有限样本性质。本文通过一个以反事实结果均值为目标的应用案例说明了该方法的使用。