  Doubly protected estimators are widely used for estimating the population
mean of an outcome Y from a sample where the response is missing in some
individuals. To compensate for the missing responses, a vector X of covariates
is observed at each individual, and the missing mechanism is assumed to be
independent of the response, conditioned on X (missing at random). In recent
years, many authors have moved from the mean to the median, and more generally,
doubly protected estimators of the quantiles have been proposed, assuming a
parametric regression model for the relationship between X and Y and a
parametric form for the propensity score. In this work, we present doubly
protected estimators for the quantiles that are also robust, in the sense that
they are resistant to the presence of outliers in the sample. We also
flexibilize the model for the relationship between X and Y . Thus we present
robust doubly protected estimators for the quantiles of the response in the
presence of missing observations, postulating a semiparametric regression model
for the relationship between the response and the covariates and a parametric
model for the propensity score.
