# High Dimensional M-Estimation with Missing Outcomes: A Semi-Parametric Framework

arXiv: https://arxiv.org/abs/1911.11345 · HTML: https://mlsanity.com/paper/1911.11345/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Nov 2019.

## Built on by (most cited first, 5 of 5)

- [Surrogate Assisted Semi-supervised Inference for High Dimensional Risk Prediction](https://mlsanity.com/paper/2105.01264/) (2021)
- [Semi-Supervised U-statistics](https://mlsanity.com/paper/2402.18921/) (2024)
- [Double Robust Semi-Supervised Inference for the Mean: Selection Bias under MAR Labeling with Decaying Overlap](https://mlsanity.com/paper/2104.06667/) (2021)
- [A Semi-Supervised Kernel Two-Sample Test](https://mlsanity.com/paper/2605.01775/) (2026)
- [Learning structured linear dynamical systems from missing observations](https://mlsanity.com/paper/2610.11869/) (2026)

## Abstract (arXiv)

We consider high dimensional M-estimation in settings where the response Y is possibly missing at random and the covariates X ∈ ℝp can be high dimensional compared to the sample size n. The parameter of interest boldsymbolθ0 ∈ ℝd is defined as the minimizer of the risk of a convex loss, under a fully non-parametric model, and boldsymbolθ0 itself is high dimensional which is a key distinction from existing works. Standard high dimensional regression and series estimation with possibly misspecified models and missing Y are included as special cases, as well as their counterparts in causal inference using 'potential outcomes'. Assuming boldsymbolθ0 is s-sparse (s ll n), we propose an L1-regularized debiased and doubly robust (DDR) estimator of boldsymbolθ0 based on a high dimensional adaptation of the traditional double robust (DR) estimator's construction. Under mild tail assumptions and arbitrarily chosen (working) models for the propensity score (PS) and the outcome regression (OR) estimators, satisfying only some high-level conditions, we establish finite sample performance bounds for the DDR estimator showing its (optimal) L2 error rate to be √s (log d)/ n when both models are correct, and its consistency and DR properties when only one of them is correct. Further, when both the models are correct, we propose a desparsified version of our DDR estimator that satisfies an asymptotic linear expansion and facilitates inference on low dimensional components of boldsymbolθ0. Finally, we discuss various of choices of high dimensional parametric/semi-parametric working models for the PS and OR estimators. All results are validated via detailed simulations.