  We derive an information criterion to select a parametric model of
complete-data distribution when only incomplete or partially observed data is
available. Compared with AIC, our new criterion has an additional penalty term
for missing data, which is expressed by the Fisher information matrices of
complete data and incomplete data. We prove that our criterion is an
asymptotically unbiased estimator of complete-data divergence, namely, the
expected Kullback-Leibler divergence between the true distribution and the
estimated distribution for complete data, whereas AIC is that for the
incomplete data. Information criteria PDIO (Shimodaira 1994) and AICcd
(Cavanaugh and Shumway 1998) have been previously proposed to estimate
complete-data divergence, and they have the same penalty term. The additional
penalty term of our criterion for missing data turns out to be only half the
value of that in PDIO and AICcd. The difference in the penalty term is
attributed to the fact that our criterion is derived under a weaker assumption.
A simulation study with the weaker assumption shows that our criterion is
unbiased while the other two criteria are biased. In addition, we review the
geometrical view of alternating minimizations of the EM algorithm. This
geometrical view plays an important role in deriving our new criterion.
