Efficient Quantile Regression Analysis With Missing Observations

Abstract

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 $\sqrt{n}$-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.

• Estimating equations
• Missing at random
• Resampling method
• Semiparametric efficient

Additional information

Notes on contributors

Xuerong Chen

Xuerong Chen, School of Statistics, Southwestern University of Finance and Economics, Chengdu, Sichuan, 611130, China (E-mail: [email protected]).

Alan T. K. Wan

Alan T. K. Wan is Professor, Department of Management Sciences, City University of Hong Kong, Hong Kong (E-mail: [email protected]).

Yong Zhou

Yong Zhou is Professor, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China, and Professor, School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, China (E-mail: [email protected]).