Missing data problems arise in many applied research studies. They may
jeopardize statistical inference of the model of interest, if the missing
mechanism is nonignorable, that is, the missing mechanism depends on the
missing values themselves even conditional on the observed data. With a
nonignorable missing mechanism, the model of interest is often not identifiable
without imposing further assumptions. We find that even if the missing
mechanism has a known parametric form, the model is not identifiable without
specifying a parametric outcome distribution. Although it is fundamental for
valid statistical inference, identifiability under nonignorable missing
mechanisms is not established for many commonly-used models. In this paper, we
first demonstrate identifiability of the normal distribution under monotone
missing mechanisms. We then extend it to the normal mixture and t mixture
models with non-monotone missing mechanisms. We discover that models under the
Logistic missing mechanism are less identifiable than those under the Probit
missing mechanism. We give necessary and sufficient conditions for
identifiability of models under the Logistic missing mechanism, which sometimes
can be checked in real data analysis. We illustrate our methods using a series
of simulations, and apply them to a real-life dataset
