  Semi-supervised (SS) inference has received much attention in recent years.
Apart from a moderate-sized labeled data, L, the SS setting is characterized by
an additional, much larger sized, unlabeled data, U. The setting of |U| >> |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 L and U to be
equally distributed, i.e., no selection bias in the labeling. Inferential
challenges in missing at random (MAR) 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 (DRSS) 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
|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.
