  Nonmonotone missing data arise routinely in empirical studies of social and
health sciences, and when ignored, can induce selection bias and loss of
efficiency. In practice, it is common to account for nonresponse under a
missing-at-random assumption which although convenient, is rarely appropriate
when nonresponse is nonmonotone. Likelihood and Bayesian missing data
methodologies often require specification of a parametric model for the full
data law, thus a priori ruling out any prospect for semiparametric inference.
In this paper, we propose an all-purpose approach which delivers semiparametric
inferences when missing data are nonmonotone and not at random. The approach is
based on a discrete choice model (DCM) as a means to generate a large class of
nonmonotone nonresponse mechanisms that are nonignorable. Sufficient conditions
for nonparametric identification are given, and a general framework for fully
parametric and semiparametric inference under an arbitrary DCM is proposed.
Special consideration is given to the case of logit discrete choice nonresponse
model (LDCM) for which we describe generalizations of inverse-probability
weighting, pattern-mixture estimation, doubly robust estimation and multiply
robust estimation.
