# Double Robust Semi-Supervised Inference for the Mean: Selection Bias under MAR Labeling with Decaying Overlap

arXiv: https://arxiv.org/abs/2104.06667 · HTML: https://mlsanity.com/paper/2104.06667/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 44 references that are in the map.
- First posted on arXiv 14 Apr 2021.

## Builds on (most cited first, 11 of 11)

- [Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression](https://mlsanity.com/paper/1804.02605/) (2018)
- [Robust Inference on Average Treatment Effects with Possibly More Covariates than Observations](https://mlsanity.com/paper/1309.4686/) (2013)
- [Semi-supervised Inference: General Theory and Estimation of Means](https://mlsanity.com/paper/1606.07268/) (2016)
- [On the role of surrogates in the efficient estimation of treatment effects with limited outcome data](https://mlsanity.com/paper/2003.12408/) (2020)
- [Efficient and Adaptive Linear Regression in Semi-Supervised Settings](https://mlsanity.com/paper/1701.04889/) (2017)
- [A unifying approach for doubly-robust ℓ1 regularized estimation of causal contrasts](https://mlsanity.com/paper/1904.03737/) (2019)
- [Semi-supervised Inference for Explained Variance in High-dimensional Linear Regression and Its Applications](https://mlsanity.com/paper/1806.06179/) (2018)
- [Nonparametric inverse probability weighted estimators based on the highly adaptive lasso](https://mlsanity.com/paper/2005.11303/) (2020)
- [Sparsity Double Robust Inference of Average Treatment Effects](https://mlsanity.com/paper/1905.00744/) (2019)
- [High Dimensional M-Estimation with Missing Outcomes: A Semi-Parametric Framework](https://mlsanity.com/paper/1911.11345/) (2019)
- [Asymptotic causal inference with observational studies trimmed by the estimated propensity scores](https://mlsanity.com/paper/1704.00666/) (2017)

## Built on by (most cited first, 4 of 4)

- [Efficient and Multiply Robust Risk Estimation under General Forms of Dataset Shift](https://mlsanity.com/paper/2306.16406/) (2023)
- [Bayesian Semi-supervised Inference via a Debiased Modeling Approach](https://mlsanity.com/paper/2509.17385/) (2025)
- [Bayesian Semiparametric Causal Inference: Targeted Doubly Robust Estimation of Treatment Effects](https://mlsanity.com/paper/2511.15904/) (2025)
- [Latent Factor Point Processes for Patient Representation in Electronic Health Records](https://mlsanity.com/paper/2508.20327/) (2025)

## Abstract (arXiv)

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.