  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.
