## The Annals of Statistics

### High dimensional censored quantile regression

#### Abstract

Censored quantile regression (CQR) has emerged as a useful regression tool for survival analysis. Some commonly used CQR methods can be characterized by stochastic integral-based estimating equations in a sequential manner across quantile levels. In this paper, we analyze CQR in a high dimensional setting where the regression functions over a continuum of quantile levels are of interest. We propose a two-step penalization procedure, which accommodates stochastic integral based estimating equations and address the challenges due to the recursive nature of the procedure. We establish the uniform convergence rates for the proposed estimators, and investigate the properties on weak convergence and variable selection. We conduct numerical studies to confirm our theoretical findings and illustrate the practical utility of our proposals.

#### Article information

Source
Ann. Statist., Volume 46, Number 1 (2018), 308-343.

Dates
Revised: January 2017
First available in Project Euclid: 22 February 2018

https://projecteuclid.org/euclid.aos/1519268432

Digital Object Identifier
doi:10.1214/17-AOS1551

Mathematical Reviews number (MathSciNet)
MR3766954

Zentralblatt MATH identifier
06865113

Subjects
Primary: 62J07: Ridge regression; shrinkage estimators
Secondary: 62H12: Estimation

#### Citation

Zheng, Qi; Peng, Limin; He, Xuming. High dimensional censored quantile regression. Ann. Statist. 46 (2018), no. 1, 308--343. doi:10.1214/17-AOS1551. https://projecteuclid.org/euclid.aos/1519268432

#### References

• Belloni, A. and Chernozhukov, V. (2011). $\ell_{1}$-penalized quantile regression in high-dimensional sparse models. Ann. Statist. 39 82–130.
• Bickel, P. J., Ritov, Y. and Tsybakov, A. B. (2009). Simultaneous analysis of lasso and Dantzig selector. Ann. Statist. 37 1705–1732.
• Bradic, J., Fan, J. and Jiang, J. (2011). Regularization for Cox’s proportional hazards model with NP-dimensionality. Ann. Statist. 39 3092–3120.
• Efron, B. (1967). The two sample problem with censored data. In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability IV 831–853. Prentice Hall, Engewood Cliffs, NJ.
• Fan, J., Fan, Y. and Barut, E. (2014). Adaptive robust variable selection. Ann. Statist. 42 324–351.
• Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assoc. 96 1348–1360.
• Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis. Wiley, New York.
• He, X. and Shao, Q.-M. (2000). On parameters of increasing dimensions. J. Multivariate Anal. 73 120–135.
• Huang, J. and Ma, S. (2010). Variable selection in the accelerated failure time model via the bridge method. Lifetime Data Anal. 16 176–195.
• Huang, J., Ma, S. and Xie, H. (2006). Regularized estimation in the accelerated failure time model with high-dimensional covariates. Biometrics 62 813–820.
• Huang, J., Ma, S. and Zhang, C.-H. (2008). Adaptive Lasso for sparse high-dimensional regression models. Statist. Sinica 18 1603–1618.
• Johnson, B. A. (2009). On lasso for censored data. Electron. J. Stat. 3 485–506.
• Koenker, R. (2005). Quantile Regression. Cambridge Univ. Press, Cambridge.
• Koenker, R. (2016). quantreg: Quantile regression. R package version 5.29.
• Koenker, R. and Bassett, G. Jr. (1978). Regression quantiles. Econometrica 46 33–50.
• Neocleous, T., Branden, K. V. and Portnoy, S. (2006). Correction to: “Censored regression quantiles” [J. Amer. Statist. Assoc. 98 (2003) 1001–1012; MR2041488] by Portnoy. J. Amer. Statist. Assoc. 101 860–861.
• Peng, L. (2012). Self-consistent estimation of censored quantile regression. J. Multivariate Anal. 105 368–379.
• Peng, L. and Huang, Y. (2008). Survival analysis with quantile regression models. J. Amer. Statist. Assoc. 103 637–649.
• Peng, L., Xu, J. and Kutner, N. (2014). Shrinkage estimation of varying covariate effects based on quantile regression. Stat. Comput. 24 853–869.
• Portnoy, S. (2003). Censored regression quantiles. J. Amer. Statist. Assoc. 98 1001–1012.
• Portnoy, S. and Lin, G. (2010). Asymptotics for censored regression quantiles. J. Nonparametr. Stat. 22 115–130.
• Powell, J. L. (1986). Censored regression quantiles. J. Econometrics 32 143–155.
• Shedden, K., Taylor, J. M. et al. (2008). Gene expression-based survival prediction in lung adenocarcinoma: A multi-site, blinded validation study. Nature Medicine 14 822–827.
• Shows, J. H., Lu, W. and Zhang, H. H. (2010). Sparse estimation and inference for censored median regression. J. Statist. Plann. Inference 140 1903–1917.
• Talagrand, M. (2005). The Generic Chaining: Upper and Lower Bounds of Stochastic Processes. Springer, Berlin.
• Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B. Stat. Methodol. 58 267–288.
• van de Geer, S. A. (2008). High-dimensional generalized linear models and the lasso. Ann. Statist. 36 614–645.
• Volgushev, S., Wagener, J. and Dette, H. (2014). Censored quantile regression processes under dependence and penalization. Electron. J. Stat. 8 2405–2447.
• Wang, H. J. and Wang, L. (2009). Locally weighted censored quantile regression. J. Amer. Statist. Assoc. 104 1117–1128.
• Wang, H. J., Zhou, J. and Li, Y. (2013). Variable selection for censored quantile regresion. Statist. Sinica 23 145–167.
• Wang, L., Wu, Y. and Li, R. (2012). Quantile regression for analyzing heterogeneity in ultra-high dimension. J. Amer. Statist. Assoc. 107 214–222.
• Ying, Z., Jung, S. H. and Wei, L. J. (1995). Survival analysis with median regression models. J. Amer. Statist. Assoc. 90 178–184.
• Zhang, C.-H. (2010). Nearly unbiased variable selection under minimax concave penalty. Ann. Statist. 38 894–942.
• Zhang, C.-H. and Huang, J. (2008). The sparsity and bias of the LASSO selection in high-dimensional linear regression. Ann. Statist. 36 1567–1594.
• Zheng, Q., Gallagher, C. and Kulasekera, K. B. (2013). Adaptive penalized quantile regression for high dimensional data. J. Statist. Plann. Inference 143 1029–1038.
• Zheng, Q., Peng, L. and He, X. (2015). Globally adaptive quantile regression with ultra-high dimensional data. Ann. Statist. 43 2225–2258.
• Zheng, Q., Peng, L. and He, X. (2018). Supplement to “High dimensional censored quantile regression.” DOI:10.1214/17-AOS1551SUPP.
• Zou, H. (2006). The adaptive lasso and its oracle properties. J. Amer. Statist. Assoc. 101 1418–1429.

#### Supplemental materials

• Supplement to “High dimensional censored quantile regression”. Additional simulation results, remarks, and proofs of technical lemmas.