Advertisement

Single index quantile regression for censored data

  • Eliana ChristouEmail author
  • Michael G. Akritas
Original Paper
  • 29 Downloads

Abstract

Quantile regression (QR) has become a popular method of data analysis, especially when the error term is heteroscedastic. It is particularly relevant for the analysis of censored survival data as an alternative to proportional hazards and the accelerated failure time models. Such data occur frequently in biostatistics, environmental sciences, social sciences and econometrics. There is a large body of work for linear/nonlinear QR models for censored data, but it is only recently that the single index quantile regression (SIQR) model has received some attention. However, the only existing method for fitting the SIQR model for censored data uses an iterative algorithm and no asymptotic theory for the resulting estimator of the parametric component is given. We propose a non-iterative estimation algorithm and derive the asymptotic distribution of the proposed estimator under heteroscedasticity. Results from simulation studies evaluating the finite sample performance of the proposed estimator are reported.

Keywords

Censored data Dimension reduction Index model Quantile regression 

Notes

Acknowledgements

The authors wish to thank the two referees, whose comments lead to improvements in the presentation of this paper.

Supplementary material

10260_2019_450_MOESM1_ESM.pdf (232 kb)
Supplementary material 1 (pdf 231 KB)

References

  1. Akritas MG (1996) On the use of nonparametric regression techniques for fitting parametric regression models. Biometrics 52(4):1342–1362MathSciNetzbMATHGoogle Scholar
  2. Bücher A, El Ghouch A, Van Keilegom I (2014) Single-index quantile regression models for censored data. Working paper. http://dial.uclouvain.be/handle/boreal:139328?site_name=UCL
  3. Christou E, Akritas MG (2016) Single index quantile regression for heteroscedastic data. J Multivar Anal 150:169–182MathSciNetzbMATHGoogle Scholar
  4. Christou E, Akritas MG (2018) Variable selection in heteroscedastic single-index quantile regression. Commun Stat Theory Methods 47(24):6019–6033MathSciNetGoogle Scholar
  5. Cox DR (1972) Regression models and life-tables. J R Stat Soc Ser B 34(2):187–220MathSciNetzbMATHGoogle Scholar
  6. Efron B (1967) The two-sample problem with censored data. In: Le Cam L, Neyman J (eds) Proceedings of the 5th Berkeley symposium in mathematical statistics, vol IV. Prentice-Hall, Upper Saddle River, pp 831–853Google Scholar
  7. Gannoun A, Saracco J, Yu K (2007) Comparison of kernel estimators of conditional distribution function and quantile regression under censoring. Stat Model 7(4):329–344MathSciNetGoogle Scholar
  8. Gonzalez-Manteiga W, Cadarso-Suarez C (1994) Asymptotic properties of a generalized Kaplan–Meier estimator with some applications. J Nonparametr Stat 4:65–78MathSciNetzbMATHGoogle Scholar
  9. Hansen B (2008) Uniform convergence rates for kernel estimation with dependent data. Econ Theory 24:726–748MathSciNetzbMATHGoogle Scholar
  10. Hjort NL, Pollard D (1993) Asymptotics for minimisers of convex processes. Unpublished manuscript. http://arxiv.org/pdf/1107.3806v1.pdf
  11. Honoré C, Khan S, Powell J (2002) Quantile regression under random censoring. J Econ 109:67–105MathSciNetzbMATHGoogle Scholar
  12. Hosmer D, Lemeshow S (1999) Applied survival analysis: regression modelling of time to evet data. Wiley, New YorkzbMATHGoogle Scholar
  13. Koenker R, Bassett G (1978) Regression quantiles. Econometrica 46(1):33–50MathSciNetzbMATHGoogle Scholar
  14. Koenker R, Geling O (2001) Reappraising medfly longevity: a quantile regression survival analysis. J Am Stat Assoc 96(454):458–468MathSciNetzbMATHGoogle Scholar
  15. Koenker R, Park BJ (1994) An interior point algorithm for nonlinear quantile regression. J Econ 71(1–2):265–283MathSciNetzbMATHGoogle Scholar
  16. Kong E, Xia Y (2012) A single-index quantile regression model and its estimation. Econ Theory 28:730–768MathSciNetzbMATHGoogle Scholar
  17. Kong E, Linton O, Xia Y (2013) Global bahadur representation for nonparametric censored regression quantiles and its applications. Econ Theory 29(05):941–968MathSciNetzbMATHGoogle Scholar
  18. Li K-C, Duan N (1989) Regression analysis under link violation. Ann Stat 17(3):1009–1052MathSciNetzbMATHGoogle Scholar
  19. Li K-C, Wang J-L, Chen C-H (1999) Dimension reduction for censored regression data. Ann Stat 27(1):1–23MathSciNetzbMATHGoogle Scholar
  20. Ma S, He X (2016) Inference for single-index quantile regression models with profile optimization. Ann Stat 44(3):1234–1268MathSciNetzbMATHGoogle Scholar
  21. Mezzetti M, Giudici P (1999) Monte Carlo methods for nonparametric survival model determination. J Ital Statist Soc 1:49–60Google Scholar
  22. Powell JL (1984) Least absolute deviations estimation for the censored regression model. J Econ 25:303–325MathSciNetzbMATHGoogle Scholar
  23. Powell JL (1986) Censored regression quantiles. J Econ 32:143–155MathSciNetzbMATHGoogle Scholar
  24. Wang HJ, Wang L (2009) Locally weighted censored quantile regression. J Am Stat Assoc 104(487):1117–1128MathSciNetzbMATHGoogle Scholar
  25. Wu TZ, Yu K, Yu Y (2010) Single index quantile regression. J Multivar Anal 101(7):1607–1621MathSciNetzbMATHGoogle Scholar
  26. Ying Z, Jung SH, Wei LJ (1995) Survival analysis with median regression models. J Am Stat Assoc 90(429):178–184MathSciNetzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of North Carolina at CharlotteCharlotteUSA
  2. 2.Department of StatisticsThe Pennsylvania State UniversityState CollegeUSA

Personalised recommendations