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：本文关注当结果随机缺失时，高维线性模型回归函数的推断问题。我们提出了一种估计量，该估计量将回归函数的Lasso初始估计与基于Lasso回归加权残差的偏差校正项相结合。权重依赖于缺失概率（倾向得分）的估计，并通过求解一个在偏差和方差之间最优权衡的凸优化程序得到。假设倾向得分可以在样本内数据点处得到逐点一致的估计，本文提出的回归函数估计量在所有渐近线性估计量中具备渐近正态性和半参数有效性。此外，即便倾向得分是通过现代机器学习技术估计的，该估计量仍能保持其渐近性质。我们通过对比模拟研究以及实际天体质量问题（斯隆数字巡天中星系恒星质量的推断）验证了所提估计量的有限样本性能。