Weighted composite quantile regression for single index model with missing covariates at random

  • Huilan LiuEmail author
  • Hu Yang
  • Changgen Peng
Original paper


This paper considers weighted composite quantile estimation of the single-index model with missing covariates at random. Under some regularity conditions, we establish the large sample properties of the estimated index parameters and link function. The large sample properties of the parametric part show that the estimator with estimated selection probability have a smaller limiting variance than the one with the true selection probability. However, the large sample properties of the estimated link function indicate that whether weights were estimated or not has no effect on the asymptotic variance. Studies of simulation and the real data analysis are presented to illustrate the behavior of the proposed estimators.


Horvitz–Thompson property Local linear regression Missing at random 



The authors sincerely thank the Editor, the Associate Editor and two Reviewers for their helpful comments and suggestions which lead to a significant improvement on this paper. Liu’s work is supported by the National Natural Science Foundation of China (Grant No. 11761020), China Postdoctoral Science Foundation(Grant No. 2017M623067), Open Foundation of Guizhou Provincial Key Laboratory of Public Big Data(Grant No. 2017BDKFJJ030), Scientific Research Foundation for Young Talents of Department of Education of Guizhou Province(Grant No. 2017104), Science and Technology Foundation of Guizhou Province (Grant No. QKH20177222). Yang’s work is supported by the National Natural Science Foundation of China (Grant No. 11671059). Peng’s work is supported by the National Natural Science Foundation of China (Grant No. 61662009), Science and Technology Foundation of Guizhou Province (Grant No. QKH20183001).


  1. Chaudhuri P, Doksum K, Samarov A (1997) On average derivative quantile regression. Ann Stat 25:715–744MathSciNetCrossRefzbMATHGoogle Scholar
  2. Guo X, Xu W, Zhu L (2014) Multi-index regression models with missing covariates at random. J Multivar Anal 123:345–363MathSciNetCrossRefzbMATHGoogle Scholar
  3. Härdle W, Stoker TM (1989) Investigating smooth multiple regression by the method of average derivatives. J Am Stat Assoc 84:986–995MathSciNetzbMATHGoogle Scholar
  4. Hjort NL, Pollard D (2011) Asymptotics for minimisers of convex processes. arXiv:1107.3806
  5. Horvitz DG, Thompson DJ (1952) A generalization of sampling without replacement from a finite universe. J Am Stat Assoc 47:663–685MathSciNetCrossRefzbMATHGoogle Scholar
  6. Jiang R, Zhou Z, Qian W, Shao W (2012) Single-index composite quantile regression. J Korean Stat Soc 41:323–332MathSciNetCrossRefzbMATHGoogle Scholar
  7. Kai B, Li R, Zou H (2010) Local composite quantile regression smoothing: an efficient and safe alternative to local polynomial regression. J R Stat Soc Ser B 72:49–69MathSciNetCrossRefGoogle Scholar
  8. Kai B, Li R, Zou H (2011) New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models. Ann Stat 39:305–332MathSciNetCrossRefzbMATHGoogle Scholar
  9. Knight K (1998) Limiting distributions for L1 regression estimators under general conditions. Ann Stat 26:755–770CrossRefzbMATHGoogle Scholar
  10. Koenker R, Bassett G (1978) Regression quantiles. Econometrica 46:33–50MathSciNetCrossRefzbMATHGoogle Scholar
  11. Kong E, Xia Y (2014) An adaptive composite quantile approach to dimension reduction. Ann Stat 42:1657–1688MathSciNetCrossRefzbMATHGoogle Scholar
  12. Li KC (1991) Sliced inverse regression for dimension reduction. J Am Stat Assoc 86:316–327MathSciNetCrossRefzbMATHGoogle Scholar
  13. Li KC (1992) On principal Hessian directions for data visualization and dimension reduction: another application of Stein’s lemma. J Am Stat Assoc 87:1025–1039MathSciNetCrossRefzbMATHGoogle Scholar
  14. Li T, Yang H (2016) Inverse probability weighted estimators for single-index models with missing covariates. Commun Stat-Theory Methods 45:1199–1214MathSciNetCrossRefzbMATHGoogle Scholar
  15. Li J, Li Y, Zhang R (2017) B spline variable selection for the single index models. Stat Pap 58:691–706MathSciNetCrossRefzbMATHGoogle Scholar
  16. Liang H (2008) Generalized partially linear models with missing covariates. J Multivar Anal 99:880–895MathSciNetCrossRefzbMATHGoogle Scholar
  17. Liang H, Wang S, Robins JM, Carroll RJ (2011) Estimation in partially linear models with missing covariates. J Am Stat Assoc 99:357–367MathSciNetCrossRefzbMATHGoogle Scholar
  18. Little RJ, Rubin DB (1987) Statistical analysis with missing data. Wiley, New YorkzbMATHGoogle Scholar
  19. Liu H, Yang H (2017) Estimation and variable selection in single-index composite quantile regression. Commun Stat-Simul Comput 46:7022–7039MathSciNetCrossRefzbMATHGoogle Scholar
  20. Liu H, Yang H, Xia X (2017) Robust estimation and variable selection in censored partially linear additive models. J Korean Stat Soc 46:88–103MathSciNetCrossRefzbMATHGoogle Scholar
  21. Lv Y, Zhang R, Zhao W, Liu J (2014) Quantile regression and variable selection for the single-index model. J Appl Stat 41:1565–1577MathSciNetCrossRefGoogle Scholar
  22. Mack YP, Silverman BW (1982) Weak and strong uniform consistency of kernel regression estimates. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 61:405–415MathSciNetCrossRefzbMATHGoogle Scholar
  23. Peng H, Huang T (2011) Penalized least squares for single index models. J Stat Plan Inference 141:1362–1379MathSciNetCrossRefzbMATHGoogle Scholar
  24. Sherwood B, Wang L, Zhou XH (2013) Weighted quantile regression for analyzing health care cost data with missing covariates. Stat Med 32:4967–4979MathSciNetCrossRefGoogle Scholar
  25. Wang CY, Wang S, Gutierrez RG, Carroll RJ (1998) Local linear regression for generalized linear models with missing data. Ann Stat 26:1028–1050MathSciNetCrossRefzbMATHGoogle Scholar
  26. Wang CY, Chen HY (2001) Augmented inverse probability weighted estimator for Cox missing covariate regression. Biometrics 57:414–419MathSciNetCrossRefzbMATHGoogle Scholar
  27. Wong H, Guo S, Chen M, Wai-Cheung IP (2009) On locally weighted estimation and hypothesis testing of varying-coefficient models with missing covariates. J Stat Plan Inference 139:2933–2951MathSciNetCrossRefzbMATHGoogle Scholar
  28. Wu T, Yu K, Yu Y (2010) Single-index quantile regression. J Multivar Anal 101:1607–1621MathSciNetCrossRefzbMATHGoogle Scholar
  29. Xia Y, Tong H, Li WK, Zhu LX (2002) An adaptive estimation of dimension reduction space. J R Stat Soc Ser B 64:363–410MathSciNetCrossRefzbMATHGoogle Scholar
  30. Xia Y, Härdle W (2006) Semi-parametric estimation of partially linear single-index models. J Multivar Anal 97:1162–1184MathSciNetCrossRefzbMATHGoogle Scholar
  31. Yang H, Liu HL (2016) Penalized weighted composite quantile estimators with missing covariates. Stat Pap 57:69–88MathSciNetCrossRefzbMATHGoogle Scholar
  32. Zou H, Yuan M (2008) Composite quantile regression and the oracle model selection theory. Ann Stat 36:1108–1126MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Guizhou Provincial Key Laboratory of Public Big DataGuizhou UniversityGuiyangPeople’s Republic of China
  2. 2.College of Mathematics and StatisticsGuizhou UniversityGuiyangPeople’s Republic of China
  3. 3.College of Mathematics and StatisticsChongqing UniversityChongqingPeople’s Republic of China

Personalised recommendations