  Statistical inference is considered for variables of interest, called primary
variables, when auxiliary variables are observed along with the primary
variables. We consider the setting of incomplete data analysis, where some
primary variables are not observed. Utilizing a parametric model of joint
distribution of primary and auxiliary variables, it is possible to improve the
estimation of parametric model for the primary variables when the auxiliary
variables are closely related to the primary variables. However, the estimation
accuracy reduces when the auxiliary variables are irrelevant to the primary
variables. For selecting useful auxiliary variables, we formulate the problem
as model selection, and propose an information criterion for predicting primary
variables by leveraging auxiliary variables. The proposed information criterion
is an asymptotically unbiased estimator of the Kullback-Leibler divergence for
complete data of primary variables under some reasonable conditions. We also
clarify an asymptotic equivalence between the proposed information criterion
and a variant of leave-one-out cross validation. Performance of our method is
demonstrated via a simulation study and a real data example.
