  We study the identification and estimation of statistical functionals of
multivariate data missing non-monotonically and not-at-random, taking a
semiparametric approach. Specifically, we assume that the missingness mechanism
satisfies what has been previously called "no self-censoring" or "itemwise
conditionally independent nonresponse," which roughly corresponds to the
assumption that no partially-observed variable directly determines its own
missingness status. We show that this assumption, combined with an odds ratio
parameterization of the joint density, enables identification of functionals of
interest, and we establish the semiparametric efficiency bound for the
nonparametric model satisfying this assumption. We propose a practical
augmented inverse probability weighted estimator, and in the setting with a
(possibly high-dimensional) always-observed subset of covariates, our proposed
estimator enjoys a certain double-robustness property. We explore the
performance of our estimator with simulation experiments and on a
previously-studied data set of HIV-positive mothers in Botswana.
