  To identify the estimand in missing data problems and observational studies,
it is common to base the statistical estimation on the "missing at random" and
"no unmeasured confounder" assumptions. However, these assumptions are
unverifiable using empirical data and pose serious threats to the validity of
the qualitative conclusions of the statistical inference. A sensitivity
analysis asks how the conclusions may change if the unverifiable assumptions
are violated to a certain degree. In this paper we consider a marginal
sensitivity model which is a natural extension of Rosenbaum's sensitivity model
that is widely used for matched observational studies. We aim to construct
confidence intervals based on inverse probability weighting estimators, such
that asymptotically the intervals have at least nominal coverage of the
estimand whenever the data generating distribution is in the collection of
marginal sensitivity models. We use a percentile bootstrap and a generalized
minimax/maximin inequality to transform this intractable problem to a linear
fractional programming problem, which can be solved very efficiently. We
illustrate our method using a real dataset to estimate the causal effect of
fish consumption on blood mercury level.
