  Causal weighted quantile treatment effects (WQTE) are a useful compliment to
standard causal contrasts that focus on the mean when interest lies at the
tails of the counterfactual distribution. To-date, however, methods for
estimation and inference regarding causal WQTEs have assumed complete data on
all relevant factors. Missing or incomplete data, however, is a widespread
challenge in practical settings, particularly when the data are not collected
for research purposes such as electronic health records and disease registries.
Furthermore, in such settings may be particularly susceptible to the outcome
data being missing-not-at-random (MNAR). In this paper, we consider the use of
double-sampling, through which the otherwise missing data is ascertained on a
sub-sample of study units, as a strategy to mitigate bias due to MNAR data in
the estimation of causal WQTEs. With the additional data in-hand, we present
identifying conditions that do not require assumptions regarding missingness in
the original data. We then propose a novel inverse-probability weighted
estimator and derive its' asymptotic properties, both pointwise at specific
quantiles and uniform across a range of quantiles in (0,1), when the propensity
score and double-sampling probabilities are estimated. For practical inference,
we develop a bootstrap method that can be used for both pointwise and uniform
inference. A simulation study is conducted to examine the finite sample
performance of the proposed estimators.

Methodology