  Classical semiparametric inference with missing outcome data is not robust to
contamination of the observed data and a single observation can have
arbitrarily large influence on estimation of a parameter of interest. This
sensitivity is exacerbated when inverse probability weighting methods are used,
which may overweight contaminated observations. We introduce inverse
probability weighted, double robust and outcome regression estimators of
location and scale parameters, which are robust to contamination in the sense
that their influence function is bounded. We give asymptotic properties and
study finite sample behaviour. Our simulated experiments show that
contamination can be more serious a threat to the quality of inference than
model misspecification. An interesting aspect of our results is that the
auxiliary outcome model used to adjust for ignorable missingness by some of the
estimators, is also useful to protect against contamination. We also illustrate
through a case study how both adjustment to ignorable missingness and
protection against contamination are achieved through weighting schemes, which
can be contrasted to gain further insights.
