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Missing data problems arise in many applied research studies. They mayjeopardize statistical inference of the model of interest, if the missingmechanism is nonignorable, that is, the missing mechanism depends on themissing values themselves even conditional on the observed data. With anonignorable missing mechanism, the model of interest is often not identifiablewithout imposing further assumptions. We find that even if the missingmechanism has a known parametric form, the model is not identifiable withoutspecifying a parametric outcome distribution. Although it is fundamental forvalid statistical inference, identifiability under nonignorable missingmechanisms is not established for many commonly-used models. In this paper, wefirst demonstrate identifiability of the normal distribution under monotonemissing mechanisms. We then extend it to the normal mixture and $t$ mixturemodels with non-monotone missing mechanisms. We discover that models under theLogistic missing mechanism are less identifiable than those under the Probitmissing mechanism. We give necessary and sufficient conditions foridentifiability of models under the Logistic missing mechanism, which sometimescan be checked in real data analysis. We illustrate our methods using a seriesof simulations, and apply them to a real-life dataset.

Identifiability of Normal and Normal Mixture Models With Nonignorable Missing Data