Abstract
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.
Highlights
Auxiliary variables are often observed along with primary variables
When the auxiliary variables have a close relation with the primary variables, we expect to improve the accuracy of predictive distribution of the primary variables by considering a joint modeling of the primary and auxiliary variables
We show an asymptotic equivalence between the proposed criterion and a variant of leave-one-out cross validation (LOOCV); this result is a generalization of the relationship between Takeuchi Information Criterion (TIC)
Summary
Auxiliary variables are often observed along with primary variables. Here, the primary variables are random variables of interest, and our purpose is to estimate their predictive distribution, i.e., a probability distribution of the primary variables in future test data, while the auxiliary variables are random variables that are observed in training data but not included in the primary variables.We assume that the auxiliary variables are not observed in the test data, or we do not use them even if they are observed in the test data. Auxiliary variables are often observed along with primary variables. The primary variables are random variables of interest, and our purpose is to estimate their predictive distribution, i.e., a probability distribution of the primary variables in future test data, while the auxiliary variables are random variables that are observed in training data but not included in the primary variables. We assume that the auxiliary variables are not observed in the test data, or we do not use them even if they are observed in the test data. The notion of auxiliary variables has been considered in statistics and machine learning literature. The “curds and whey” method [1] and the “coaching variables” method [2] are based on a similar idea for improving prediction accuracy of primary variables by using auxiliary variables
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