Annals of the Institute of Statistical Mathematics

, Volume 72, Issue 1, pp 265–295

# Semiparametric quantile regression with random censoring

Article

## Abstract

This paper considers estimation and inference in semiparametric quantile regression models when the response variable is subject to random censoring. The paper considers both the cases of independent and dependent censoring and proposes three iterative estimators based on inverse probability weighting, where the weights are estimated from the censoring distribution using the Kaplan–Meier, a fully parametric and the conditional Kaplan–Meier estimators. The paper proposes a computationally simple resampling technique that can be used to approximate the finite sample distribution of the parametric estimator. The paper also considers inference for both the parametric and nonparametric components of the quantile regression model. Monte Carlo simulations show that the proposed estimators and test statistics have good finite sample properties. Finally, the paper contains a real data application, which illustrates the usefulness of the proposed methods.

## Keywords

Inverse probability of censoring Local linear estimation M-M algorithm

## Supplementary material

10463_2018_688_MOESM1_ESM.pdf (230 kb)
Supplementary material 1 (pdf 230 KB)

## References

1. Bang, H., Tsiatis, A. (2000). Estimating medical costs with censored data. Biometrika, 87, 329–343.
2. Bang, H., Tsiatis, A. (2002). Median regression with censored cost data. Biometrics, 58, 643–649.
3. Bassett, G., Koenker, R. (1978). Asymptotic theory of least absolute error regression. Journal of the American Statistical Association, 73, 618–622.
4. Beran, R. (1981). Nonparametric regression with randomly censored survival data, Technical Report, University of California, Berkeley.Google Scholar
5. Breslow, N. (1972). Discussion of a paper by D. R. Cox. Journal of the Royal Statistical Society, 34, 261–217.
6. Cai, Z., Xiao, Z. (2012). Semiparametric quantile regression estimation in dynamic models with partially varying coefficients. Journal of Econometrics, 167, 413–425.
7. Cai, Z., Xu, X. (2008). Nonparametric quantile estimations for dynamic smooth coefficient models. Journal of the American Statistical Association, 103, 1596–1608.
8. Chauduri, P. (1991). Global nonparametric estimation of conditional quantile functions and their derivatives. Journal of Multivariate Analysis, 39, 246–269.
9. Chauduri, P., Doksum, K., Samarov, A. (1997). On average derivative quantile regression. Annals of Statistics, 25, 715–744.Google Scholar
10. Cox, D. (1972). Regression models and life-tables (with discussion). Journal of the Royal Statistical Society, 34, 187–220.
11. Cox, D. (1975). Partial likelihood. Biometrika, 62, 269–272.
12. De Backer, M., El Ghouch, A., van Keilegom, I. (2017). Semiparametric copula quantile regression for complete or censored data. Electronic Journal of Statistics, 11, 1660–1698.
13. El Ghouch, A., van Keilegom, I. (2009). Local linear quantile regression with dependent censored data. Statistica Sinica, 19, 1621–1640.Google Scholar
14. Fan, J., Gijbels, I. (1994). Censored regression: Local linear approximations and their applications. Journal of the American Statistical Association, 89, 560–569.
15. Fan, J., Hu, T., Truong, Y. (1994). Robust non-parametric function estimation. Scandinavian Journal of Statistics, 21, 433–446.Google Scholar
16. Fan, J., Huang, T. (2005). Profile likelihood inferences on semiparametric varying-coefficient partially linear models. Bernoulli, 11, 1031–1057.
17. He, X., Liang, H. (2000). Quantile regression estimates for a class of linear and partially linear errors-in-variables models. Statistica Sinica, 10, 129–140.Google Scholar
18. He, X., Shi, P. (1996). Bivariate tensor product b-splines in a partially linear regression. Journal of Multivariate Analysis, 58, 162–181.
19. Horowitz, J., Lee, S. (2005). Nonparametric estimation of an additive quantile regression model. Journal of the American Statistical Association, 100, 1238–1249.
20. Hunter, D., Lange, K. (2000). Quantile regression via an MM algorithm. Journal of Computational and Graphical Statistics, 9, 60–77.
21. Jin, Z., Ying, Z., Wei, L. (2001). A simple resampling method by perturbing the minimand. Biometrika, 88, 381–390.
22. Kalbfleisch, J., Prentice, R. (2002). The statistical analysis of failure data. New York: Wiley.Google Scholar
23. Koenker, R. (2005). Quantile regression. Cambridge: Cambridge University Press.
24. Koenker, R., Bassett, G. (1978). Regression quantiles. Econometrica, 46, 33–50.
25. Koul, H., Susarla, V., Ryzin, J. V. (1981). Regression analysis of randomly right censored data. Annals of Statistics, 9, 1276–1288.Google Scholar
26. Lee, S. (2003). Efficient semiparametric estimation of a partially linear quantile regression model. Econometric Theory, 19, 1–31.
27. Leng, C., Tong, X. (2013). A quantile regression estimator for censored data. Bernoulli, 19, 344–361.
28. Li, G., Datta, S. (2001). A bootstrap approach to nonparametric regression for right censored data. Annals of the Institute of Statistical Mathematics, 53, 708–729.Google Scholar
29. Li, W., Patilea, V. (2017). A dimension reduction approach for conditional Kaplan–Meier estimators, Test forthcoming.Google Scholar
30. Lin, Y. (2000). Linear regression analysis of censored medical costs. Biostatistics, 1, 35–47.
31. Peng, L., Huang, Y. (2008). Survival analysis with quantile regression models. Journal of the American Statistical Association, 103, 637–649.
32. Robins, J., Rotnitzky, A. (1992). Recovery information and adjustment for dependent censoring using surrogate markers in AIDS epidemiology-methodological issues (pp. 297–331). Boston: Birkhauser.
33. Satten, G., Datta, S. (2001). The Kaplan–Meier estimator as an inverse-probability-of-censoring weighted average. American Statistician, 55, 207–210.
34. Su, J., Wei, L. (1991). A lack of fit test for the mean function in a generalized linear model. Journal of the American Statistical Association, 86, 420–426.
35. Van Keilegom, I., Akritas, M., Veraverbeke, N. (2001). Estimation of the conditional distribution in regression with censored data: A comparative study. Computational Statistics and Data Analysis, 35, 487–501.
36. Wang, H., Wang, L. (2009). Locally weighted censored quantile regression. Journal of the American Statistical Association, 104, 1117–1128.
37. Xie, S., Wan, A., Zhou, Y. (2015). Quantile regression methods with varying-coefficient models for censored data. Computational Statistics and Data Analysis88, 154–172.
38. Ying, Z., Jung, S., Wei, L. (1995). Survival analysis with median regression models. Journal of the American Statistical Association, 90, 178–184.
39. Yu, K., Jones, M. (1998). Local linear quantile regression. Journal of the American Statistical Association, 93, 228–237.
40. Zhou, L. (2006). A simple censored median regression estimator. Statistica Sinica, 16, 1043–1058.