We study semi-parametric estimation of the population mean when data is observed missing at random (MAR) in the $n < p$ "inconsistency regime", in which neither the outcome model nor the propensity/missingness model can be estimated consistently. Consider a high-dimensional linear-GLM specification in which the number of confounders is proportional to the sample size. In the case $n > p$, past work has developed theory for the classical AIPW estimator in this model and established its variance inflation and asymptotic normality when the outcome model is fit by ordinary least squares. Ordinary least squares is no longer feasible in the case $n < p$ studied here, and we also demonstrate that a number of classical debiasing procedures become inconsistent. This challenge motivates our development and analysis of a novel procedure: we establish that it is consistent for the population mean under proportional asymptotics allowing for $n < p$, and also provide confidence intervals for the linear model coefficients. Providing such guarantees in the inconsistency regime requires a new debiasing approach that combines penalized M-estimates of both the outcome and propensity/missingness models in a non-standard way.

翻译：我们研究当数据以随机缺失（MAR）方式观测时，在$n < p$的“不一致性机制”下总体均值的半参数估计问题，在此情形下，结果模型和倾向性/缺失模型均无法被一致估计。考虑一个高维线性广义线性模型设定，其中混杂变量数量与样本量成比例。在$n > p$情形下，前人工作已发展出该模型下经典AIPW估计量的理论，并证明了当结果模型通过普通最小二乘法拟合时，其方差膨胀与渐近正态性。然而，普通最小二乘法在我们研究的$n < p$情形下不再可行，同时我们证明若干经典去偏方法会变得不一致。这一挑战促使我们发展并分析一种新方法：我们证明在允许$n < p$的比例渐近框架下，该方法对总体均值具有一致性，并同时为线性模型系数提供置信区间。在不一致性机制下提供此类保证需要一种新颖的去偏方法，该方法以非标准方式结合了结果模型与倾向性/缺失模型的惩罚M估计。