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.

翻译：本文研究在高维线性模型中，当结果随机缺失时对回归函数进行推断的问题。我们提出了一种估计器，该估计器将回归函数的Lasso初步估计与基于Lasso回归加权残差的偏差校正项相结合。权重取决于缺失概率（倾向得分）的估计值，并通过求解一个凸优化程序来最优地权衡偏差与方差。只要倾向得分能够在样本内数据点上实现逐点一致估计，我们所提出的回归函数估计器就具有渐近正态性，并且在所有渐近线性估计器中达到半参数有效性。此外，即使倾向得分是通过现代机器学习技术估计的，所提出的估计器仍能保持其渐近性质。我们通过对比模拟研究以及斯隆数字巡天中推断星系恒星质量的实际问题，验证了所提出估计器在有限样本下的性能。