On semi-supervised estimation using exponential tilt mixture models
Ye Tian Xinwei Zhang Zhiqiang Tan
Consider a semi-supervised setting with a labeled dataset of binary responses
and predictors and an unlabeled dataset with only the predictors. Logistic
regression is equivalent to an exponential tilt model in the labeled
population. For semi-supervised estimation, we develop further analysis and
understanding of a statistical approach using exponential tilt mixture (ETM)
models and maximum nonparametric likelihood estimation, while allowing that the
class proportions may differ between the unlabeled and labeled data. We derive
asymptotic properties of ETM-based estimation and demonstrate improved
efficiency over supervised logistic regression in a random sampling setup and
an outcome-stratified sampling setup previously used. Moreover, we reconcile
such efficiency improvement with the existing semiparametric efficiency theory
when the class proportions in the unlabeled and labeled data are restricted to
be the same. We also provide a simulation study to numerically illustrate our
theoretical findings.