  Conducting valid statistical analyses is challenging in the presence of
missing-not-at-random (MNAR) data, where the missingness mechanism is dependent
on the missing values themselves even conditioned on the observed data. Here,
we consider a MNAR model that generalizes several prior popular MNAR models in
two ways: first, it is less restrictive in terms of statistical independence
assumptions imposed on the underlying joint data distribution, and second, it
allows for all variables in the observed sample to have missing values. This
MNAR model corresponds to a so-called criss-cross structure considered in the
literature on graphical models of missing data that prevents nonparametric
identification of the entire missing data model. Nonetheless, part of the
complete-data distribution remains nonparametrically identifiable. By
exploiting this fact and considering a rich class of exponential family
distributions, we establish sufficient conditions for identification of the
complete-data distribution as well as the entire missingness mechanism. We then
propose methods for testing the independence restrictions encoded in such
models using odds ratio as our parameter of interest. We adopt two
semiparametric approaches for estimating the odds ratio parameter and establish
the corresponding asymptotic theories: one involves maximizing a conditional
likelihood with order statistics and the other uses estimating equations. The
utility of our methods is illustrated via simulation studies.
