  We consider the problem of estimating the mean of a random variable Y subject
to non-ignorable missingness, i.e., where the missingness mechanism depends on
Y . We connect the auxiliary proxy variable framework for non-ignorable
missingness (West and Little, 2013) to the label shift setting (Saerens et al.,
2002). Exploiting this connection, we construct an estimator for non-ignorable
missing data that uses high-dimensional covariates (or proxies) without the
need for a generative model. In synthetic and semi-synthetic experiments, we
study the behavior of the proposed estimator, comparing it to commonly used
ignorable estimators in both well-specified and misspecified settings.
Additionally, we develop a score to assess how consistent the data are with the
label shift assumption. We use our approach to estimate disease prevalence
using a large health survey, comparing ignorable and non-ignorable approaches.
We show that failing to account for non-ignorable missingness can have profound
consequences on conclusions drawn from non-representative samples.
