## Quantile calculus and censored regression.(English)Zbl 1189.62071

Summary: 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 a fundamental and general quantile calculus on a cumulative probability scale, 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.

### MSC:

 62G08 Nonparametric regression and quantile regression 65C60 Computational problems in statistics (MSC2010) 62G20 Asymptotic properties of nonparametric inference 62N01 Censored data models 62N02 Estimation in survival analysis and censored data 62P10 Applications of statistics to biology and medical sciences; meta analysis

quantreg
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### References:

 [1] 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. · Zbl 0445.62095 [2] Buckley, J. and James, I. (1979). Linear regression with censored data. Biometrika 66 429-436. · Zbl 0425.62051 [3] Chernozhukov, V. (2005). Extremal quantile regression. Ann. Statist. 33 806-839. · Zbl 1068.62063 [4] 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. [5] Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis . Wiley, New York. · Zbl 0727.62096 [6] Gill, P. E., Murray, W. and Wright, M. H. (1991). Numerical Linear Algebra and Optimization . Addison-Wesley, Redwood City, CA. · Zbl 0714.65063 [7] 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. · Zbl 0718.60087 [8] Gutenbrunner, C. and Jurečková, J. (1992). Regression rank scores and regression quantiles. Ann. Statist. 20 305-330. · Zbl 0759.62015 [9] Honoré, B., Khan, S. and Powell, J. L. (2002). Quantile regression under random censoring. J. Econometrics 109 67-105. · Zbl 1044.62106 [10] Jin, Z., Ying, Z. and Wei, L. J. (2001). A simple resampling method by perturbing the minimand. Biometrika 88 381-390. JSTOR: · Zbl 0984.62033 [11] Koenker, R. (2008). Censored quantile regression redux. J. Stat. Softw. 27 1-25. [12] Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica 46 33-50. JSTOR: · Zbl 0373.62038 [13] Koenker, R. and D’Orey, V. (1987). Computing regression quantiles. Appl. Statist. 36 383-393. [14] Koenker, R. and Geling, O. (2001). Reappraising medfly longevity: A quantile regression survival analysis. J. Amer. Statist. Assoc. 96 458-468. JSTOR: · Zbl 1019.62100 [15] Kosorok, M. R. (2008). Introduction to Empirical Processes and Semiparametric Inference . Springer, New York. · Zbl 1180.62137 [16] Neocleous, T., Vanden Branden, K. and Portnoy, S. (2006). Correction to “Censored regression quantiles,” by S. Portnoy. J. Amer. Statist. Assoc. 101 860-861. [17] Owen, A. (1990). Empirical likelihood ratio confidence regions. Ann. Statist. 18 90-120. · Zbl 0712.62040 [18] Peng, L. and Huang, Y. (2007). Survival analysis with temporal covariate effects. Biometrika 94 719-733. · Zbl 1135.62080 [19] Peng, L. and Huang, Y. (2008). Survival analysis with quantile regression models. J. Amer. Statist. Assoc. 103 637-649. · Zbl 1408.62159 [20] Portnoy, S. (2003). Censored regression quantiles. J. Amer. Statist. Assoc. 98 1001-1012. · Zbl 1045.62099 [21] Portnoy, S. and Jurečková, J. (1999). On extreme regression quantiles. Extremes 2 227-243. · Zbl 0959.62047 [22] Powell, J. L. (1984). Least absolute deviations estimation for the censored regression model. J. Econometrics 25 303-325. · Zbl 0571.62100 [23] Powell, J. L. (1986). Censored regression quantiles. J. Econometrics 32 143-155. · Zbl 0605.62139 [24] 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. [25] Smith, R. (1994). Nonregular regression. Biometrika 81 173-183. JSTOR: · Zbl 0803.62056 [26] 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. · Zbl 1117.62435 [27] Tsiatis, A. A. (1990). Estimating regression parameters using linear rank tests for censored data. Ann. Statist. 18 354-372. · Zbl 0701.62051 [28] Wang, H. J. and Wang, L. (2009). Locally weighed censored quantile regression. J. Amer. Statist. Assoc. 104 1117-1128. · Zbl 1388.62289 [29] Ying, Z., Jung, S. H. and Wei, L. J. (1995). Survival analysis with median regression models. J. Amer. Statist. Assoc. 90 178-184. JSTOR: · Zbl 0818.62103
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