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Abstract Semi-supervised (SS) inference has received much attention in recent years. Apart from a moderate-sized labeled data, $$\mathcal L$$, the SS setting is characterized by an additional, much larger sized, unlabeled data, $$\mathcal U$$. The setting of $$|\mathcal U\ |\gg |\mathcal L\ |$$, makes SS inference unique and different from the standard missing data problems, owing to natural violation of the so-called ‘positivity’ or ‘overlap’ assumption. However, most of the SS literature implicitly assumes $$\mathcal L$$ and $$\mathcal U$$ to be equally distributed, i.e., no selection bias in the labeling. Inferential challenges in missing at random type labeling allowing for selection bias, are inevitably exacerbated by the decaying nature of the propensity score (PS). We address this gap for a prototype problem, the estimation of the response’s mean. We propose a double robust SS mean estimator and give a complete characterization of its asymptotic properties. The proposed estimator is consistent as long as either the outcome or the PS model is correctly specified. When both models are correctly specified, we provide inference results with a non-standard consistency rate that depends on the smaller size $$|\mathcal L\ |$$. The results are also extended to causal inference with imbalanced treatment groups. Further, we provide several novel choices of models and estimators of the decaying PS, including a novel offset logistic model and a stratified labeling model. We present their properties under both high- and low-dimensional settings. These may be of independent interest. Lastly, we present extensive simulations and also a real data application. 

Summary A fundamental challenge in semi-supervised learning lies in the observed data’s disproportional size when compared with the size of the data collected with missing outcomes. An implicit understanding is that the dataset with missing outcomes, being significantly larger, ought to improve estimation and inference. However, it is unclear to what extent this is correct. We illustrate one clear benefit: root-$$n$$ inference of the outcome’s mean is possible while only requiring a consistent estimation of the outcome, possibly at a rate slower than root $$n$$. This is achieved by a novel $$k$$-fold, cross-fitted, double robust estimator. We discuss both linear and nonlinear outcomes. Such an estimator is particularly suited for models that naturally do not admit root-$$n$$ consistency, such as high-dimensional, nonparametric or semiparametric models. We apply our methods to estimating heterogeneous treatment effects. 

Random forests are powerful non-parametric regression method but are severely limited in their usage in the presence of randomly censored observations, and naively applied can exhibit poor predictive performance due to the incurred biases. Based on a local adaptive representation of random forests, we develop its regression adjustment for randomly censored regression quantile models. Regression adjustment is based on a new estimating equation that adapts to censoring and leads to quantile score whenever the data do not exhibit censoring. The proposed procedure named censored quantile regression forest, allows us to estimate quantiles of time-to-event without any parametric modeling assumption. We establish its consistency under mild model specifications. Numerical studies showcase a clear advantage of the proposed procedure.