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

Understand

We consider high dimensional $M$-estimation in settings where the response $Y$ is possibly missing at random and the covariates $\mathbf{X} \in \mathbb{R}^p$ can be high dimensional compared to the sample size $n$.

• The parameter of interest $\boldsymbol{\theta}_0 \in \mathbb{R}^d$ is defined as the minimizer of the risk of a convex loss, under a fully non-parametric model, and $\boldsymbol{\theta}_0$ 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 $\boldsymbol{\theta}_0$ is $s$-sparse ($s \ll n$), we propose an $L_1$-regularized debiased and doubly robust (DDR) estimator of $\boldsymbol{\theta}_0$ based on a high dimensional adaptation of the traditional double robust (DR) estimator's construction.