Efficient Quantile Regression Analysis With Missing Observations

This article examines the problem of estimation in a quantile regression model when observations are missing at random under independent and nonidentically distributed errors. We consider three approaches of handling this problem based on nonparametric inverse probability weighting, estimating equations projection, and a combination of both. An important distinguishing feature of our methods is their ability to handle missing response and/or partially missing covariates, whereas existing techniques can handle only one or the other, but not both. We prove that our methods yield asymptotically equivalent estimators that achieve the desirable asymptotic properties of unbiasedness, normality, and -consistency. Because we do not assume that the errors are identically distributed, our theoretical results are valid under heteroscedasticity, a particularly strong feature of our methods. Under the special case of identical error distributions, all of our proposed estimators achieve the semiparametric efficiency bound. To facilitate the practical implementation of these methods, we develop an iterative method based on the majorize/minimize algorithm for computing the quantile regression estimates, and a bootstrap method for computing their variances. Our simulation findings suggest that all three methods have good finite sample properties. We further illustrate these methods by a real data example. Supplementary materials for this article are available online.

本文针对独立且非同分布误差下、观测值随机缺失的分位数回归模型（quantile regression model）中的估计问题展开研究。本文考虑了三种基于非参数逆概率加权法（nonparametric inverse probability weighting）、估计方程投影法（estimating equations projection）以及二者结合的解决方案。本方法的一项重要独特优势在于，其可同时处理缺失响应变量与/或部分缺失协变量的情形，而现有技术仅能处理其中一类问题，无法兼顾二者。本文证明，所提方法可得到渐近等价的估计量，且这些估计量具备无偏性、正态性以及n相合性（n-consistency）这些理想的渐近性质。由于我们未假设误差服从同分布，因此理论结果在异方差（heteroscedasticity）情形下依然成立，这也是本方法的一项突出优势。在误差服从同分布的特殊情形下，所有所提估计量均可达到半参数效率界（semiparametric efficiency bound）。为便于这些方法的实际应用，我们提出了基于最大化最小化（majorize/minimize）算法的迭代方法以求解分位数回归估计量，并提出了自助法（bootstrap）以计算其方差。模拟实验结果表明，三种方法均具备良好的有限样本性质。我们进一步通过真实数据集示例对所提方法进行了演示。本文的补充材料可在线获取。