  In real-world scenarios, data collection limitations often result in
partially labeled datasets, leading to difficulties in drawing reliable causal
inferences. Traditional approaches in the semi-supervised (SS) and missing data
literature may not adequately handle these complexities, leading to biased
estimates. To address these challenges, our paper introduces a novel decaying
missing-at-random (decaying MAR) framework. This framework tackles missing
outcomes in high-dimensional settings and accounts for selection bias arising
from the dependence of labeling probability on covariates. Notably, we relax
the need for a positivity condition, commonly required in the missing data
literature, and allow uniform decay of labeling propensity scores with sample
size, accommodating faster growth of unlabeled data. Our decaying MAR framework
enables easy rate double-robust (DR) estimation of average treatment effects,
succeeding where other methods fail, even with correctly specified nuisance
models. Additionally, it facilitates asymptotic normality under model
misspecification. To achieve this, we propose adaptive new targeted
bias-reducing nuisance estimators and asymmetric cross-fitting, along with a
novel semi-parametric approach that fully leverages large volumes of unlabeled
data. Our approach requires weak sparsity conditions. Numerical results confirm
our estimators' efficacy and versatility, addressing selection bias and model
misspecification.
