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.

在半监督学习场景中，我们提出了使用指数倾斜混合模型和最大非参数似然估计法进行的统计方法的进一步分析和理解，并在允许未标记和标记数据之间类别比例不同的情况下，得出了基于指数倾斜混合模型的估计的渐近性质；同时在随机抽样和结果分层抽样设置中，展示了相对于监督逻辑回归的提高效率；此外，当未标记和标记数据的类别比例受限于相同的情况下，我们以现有的半参数效率理论为支撑，对于这种效率改进做出了解释。我们还通过模拟研究来数值上验证了我们的理论发现。