  In a missing-data setting, we have a sample in which a vector of explanatory
variables x_i is observed for every subject i, while scalar outcomes y_i are
missing by happenstance on some individuals. In this work we propose robust
estimates of the distribution of the responses assuming missing at random (MAR)
data, under a semiparametric regression model. Our approach allows the
consistent estimation of any weakly continuous functional of the response's
distribution. In particular, strongly consistent estimates of any continuous
location functional, such as the median or MM functionals, are proposed. A
robust fit for the regression model combined with the robust properties of the
location functional gives rise to a robust recipe for estimating the location
parameter. Robustness is quantified through the breakdown point of the proposed
procedure. The asymptotic distribution of the location estimates is also
derived.
