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We study semi-parametric estimation of the population mean when data isobserved missing at random (MAR) in the $n < p$ "inconsistency regime", inwhich neither the outcome model nor the propensity/missingness model can beestimated consistently. Consider a high-dimensional linear-GLM specification inwhich 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 inthis model and established its variance inflation and asymptotic normality whenthe outcome model is fit by ordinary least squares. Ordinary least squares isno longer feasible in the case $n < p$ studied here, and we also demonstratethat a number of classical debiasing procedures become inconsistent. Thischallenge motivates our development and analysis of a novel procedure: weestablish that it is consistent for the population mean under proportionalasymptotics allowing for $n < p$, and also provide confidence intervals for thelinear model coefficients. Providing such guarantees in the inconsistencyregime requires a new debiasing approach that combines penalized M-estimates ofboth the outcome and propensity/missingness models in a non-standard way.

Challenges of the inconsistency regime: Novel debiasing methods for missing data models