  This paper is concerned with inference on the regression function of a
high-dimensional linear model when outcomes are missing at random. We propose
an estimator which combines a Lasso pilot estimate of the regression function
with a bias correction term based on the weighted residuals of the Lasso
regression. The weights depend on estimates of the missingness probabilities
(propensity scores) and solve a convex optimization program that trades off
bias and variance optimally. Provided that the propensity scores can be
pointwise consistently estimated at in-sample data points, our proposed
estimator for the regression function is asymptotically normal and
semi-parametrically efficient among all asymptotically linear estimators.
Furthermore, the proposed estimator keeps its asymptotic properties even if the
propensity scores are estimated by modern machine learning techniques. We
validate the finite-sample performance of the proposed estimator through
comparative simulation studies and the real-world problem of inferring the
stellar masses of galaxies in the Sloan Digital Sky Survey.
