The Annals of Statistics

Quantile calculus and censored regression

Yijian Huang

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Quantile regression has been advocated in survival analysis to assess evolving covariate effects. However, challenges arise when the censoring time is not always observed and may be covariate-dependent, particularly in the presence of continuously-distributed covariates. In spite of several recent advances, existing methods either involve algorithmic complications or impose a probability grid. The former leads to difficulties in the implementation and asymptotics, whereas the latter introduces undesirable grid dependence. To resolve these issues, we develop fundamental and general quantile calculus on cumulative probability scale in this article, upon recognizing that probability and time scales do not always have a one-to-one mapping given a survival distribution. These results give rise to a novel estimation procedure for censored quantile regression, based on estimating integral equations. A numerically reliable and efficient Progressive Localized Minimization (PLMIN) algorithm is proposed for the computation. This procedure reduces exactly to the Kaplan–Meier method in the k-sample problem, and to standard uncensored quantile regression in the absence of censoring. Under regularity conditions, the proposed quantile coefficient estimator is uniformly consistent and converges weakly to a Gaussian process. Simulations show good statistical and algorithmic performance. The proposal is illustrated in the application to a clinical study.

Article information

Ann. Statist. Volume 38, Number 3 (2010), 1607-1637.

First available in Project Euclid: 24 March 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62N02: Estimation
Secondary: 62N01: Censored data models

Differential equation estimating integral equation piecewise-linear programming PLMIN algorithm quantile equality fraction regression quantile relative quantile varying-coefficient model


Huang, Yijian. Quantile calculus and censored regression. Ann. Statist. 38 (2010), no. 3, 1607--1637. doi:10.1214/09-AOS771.

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  • Aalen, O. O. (1980). A model for nonparametric regression analysis of counting processes. In Mathematical Statistics and Probability Theory (Proc. Sixth Internat. Conf., Wisła, 1978). (W. Klonecki, A. Kozek and J. Rosiński, eds.). Lecture Notes in Statist. 2 1–25. Springer, New York.
  • Buckley, J. and James, I. (1979). Linear regression with censored data. Biometrika 66 429–436.
  • Chernozhukov, V. (2005). Extremal quantile regression. Ann. Statist. 33 806–839.
  • Efron, B. (1967). The two sample problem with censored data. In Proc. Fifth Berkeley Symp. Math. Statist. Probab. 4 831–853. Prentice Hall, New York.
  • Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis. Wiley, New York.
  • Gill, P. E., Murray, W. and Wright, M. H. (1991). Numerical Linear Algebra and Optimization. Addison-Wesley, Redwood City, CA.
  • Gill, R. D. and Johansen, S. (1990). A survey of product-integration with a view towards application in survival analysis. Ann. Statist. 18 1501–1555.
  • Gutenbrunner, C. and Jurečková, J. (1992). Regression rank scores and regression quantiles. Ann. Statist. 20 305–330.
  • Honoré, B., Khan, S. and Powell, J. L. (2002). Quantile regression under random censoring. J. Econometrics 109 67–105.
  • Jin, Z., Ying, Z. and Wei, L. J. (2001). A simple resampling method by perturbing the minimand. Biometrika 88 381–390.
  • Koenker, R. (2008). Censored quantile regression redux. J. Stat. Softw. 27 1–25.
  • Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica 46 33–50.
  • Koenker, R. and D’Orey, V. (1987). Computing regression quantiles. Appl. Statist. 36 383–393.
  • Koenker, R. and Geling, O. (2001). Reappraising medfly longevity: A quantile regression survival analysis. J. Amer. Statist. Assoc. 96 458–468.
  • Kosorok, M. R. (2008). Introduction to Empirical Processes and Semiparametric Inference. Springer, New York.
  • Neocleous, T., Vanden Branden, K. and Portnoy, S. (2006). Correction to “Censored regression quantiles,” by S. Portnoy. J. Amer. Statist. Assoc. 101 860–861.
  • Owen, A. (1990). Empirical likelihood ratio confidence regions. Ann. Statist. 18 90–120.
  • Peng, L. and Huang, Y. (2007). Survival analysis with temporal covariate effects. Biometrika 94 719–733.
  • Peng, L. and Huang, Y. (2008). Survival analysis with quantile regression models. J. Amer. Statist. Assoc. 103 637–649.
  • Portnoy, S. (2003). Censored regression quantiles. J. Amer. Statist. Assoc. 98 1001–1012.
  • Portnoy, S. and Jurečková, J. (1999). On extreme regression quantiles. Extremes 2 227–243.
  • Powell, J. L. (1984). Least absolute deviations estimation for the censored regression model. J. Econometrics 25 303–325.
  • Powell, J. L. (1986). Censored regression quantiles. J. Econometrics 32 143–155.
  • Robins, J. M. and Ritov, Y. (1997). Toward a curse of dimensionality appropriate (CODA) asymptotic theory for semi-parametric models. Stat. Med. 16 285–319.
  • Smith, R. (1994). Nonregular regression. Biometrika 81 173–183.
  • Tian, L., Zucker, D. and Wei, L. J. (2005). On the Cox model with time-varying regression coefficients. J. Amer. Statist. Assoc. 100 172–183.
  • Tsiatis, A. A. (1990). Estimating regression parameters using linear rank tests for censored data. Ann. Statist. 18 354–372.
  • Wang, H. J. and Wang, L. (2009). Locally weighed censored quantile regression. J. Amer. Statist. Assoc. 104 1117–1128.
  • Ying, Z., Jung, S. H. and Wei, L. J. (1995). Survival analysis with median regression models. J. Amer. Statist. Assoc. 90 178–184.