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Volgushev, Stanislav; Dette, Holger

Nonparametric quantile regression for twice censored data. (English) Zbl 1273.62092

Bernoulli 19, No. 3, 748-779 (2013).
Summary: We consider the problem of nonparametric quantile regression for twice censored data. Two new estimates are presented, which are constructed by applying concepts of monotone rearrangements to estimates of the conditional distribution function. The proposed methods avoid the problem of crossing quantile curves. Weak uniform consistency and weak convergence is established for both estimates and their finite sample properties are investigated by means of a simulation study. As a by-product, we obtain a new result regarding the weak convergence of the R. Beran [Nonparametric regression with randomly censored survival data. Tech. Rep., Univ. California, Berkeley (1981)] estimator for right censored data on the maximal possible domain, which is of its own interest.

Cited in 12 Documents

MSC:

62G08 Nonparametric regression and quantile regression
62N01 Censored data models
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)

Keywords:

Beran estimator; crossing quantile curves; monotone rearrangements; survival analysis

Software:

SPLIDA; quantreg
PDFBibTeX XMLCite
\textit{S. Volgushev} and \textit{H. Dette}, Bernoulli 19, No. 3, 748--779 (2013; Zbl 1273.62092)
Full Text: DOI arXiv Euclid

References:

[1] Abberger, K. (2001). Cross validation in nonparametric quantile regression. Allg. Statist. Archiv. 82 149-161.
[2] Andersen, P.K., Borgan, Ø., Gill, R.D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer Series in Statistics . New York: Springer. · Zbl 0769.62061
[3] Anevski, D. and Fougères, A.L. (2007). Limit properties of the monotone rearrangement for density and regression function estimation. Available at . 0710.4617v1
[4] Bennett, C. and Sharpley, R. (1988). Interpolation of Operators. Pure and Applied Mathematics 129 . Boston, MA: Academic Press. · Zbl 0647.46057
[5] Beran, R. (1981). Nonparametric regression with randomly censored survival data. Technical report, Univ. California, Berkeley.
[6] Chang, M.N. (1990). Weak convergence of a self-consistent estimator of the survival function with doubly censored data. Ann. Statist. 18 391-404. · Zbl 0706.62044 · doi:10.1214/aos/1176347506
[7] Chang, M.N. and Yang, G.L. (1987). Strong consistency of a nonparametric estimator of the survival function with doubly censored data. Ann. Statist. 15 1536-1547. · Zbl 0629.62040 · doi:10.1214/aos/1176350608
[8] Chaudhuri, P. (1991). Nonparametric estimates of regression quantiles and their local Bahadur representation. Ann. Statist. 19 760-777. · Zbl 0728.62042 · doi:10.1214/aos/1176348119
[9] Chernozhukov, V., Fernandéz-Val, I. and Galichon, A. (2010). Quantile and probability curves without crossing. Econometrica 78 1093-1125. · Zbl 1192.62255 · doi:10.3982/ECTA7880
[10] Dabrowska, D.M. (1987). Nonparametric regression with censored survival time data. Scand. J. Stat. 14 181-197. · Zbl 0641.62024
[11] Dabrowska, D.M. (1989). Uniform consistency of the kernel conditional Kaplan-Meier estimate. Ann. Statist. 17 1157-1167. · Zbl 0687.62035 · doi:10.1214/aos/1176347261
[12] Dabrowska, D.M. (1992). Nonparametric quantile regression with censored data. Sankhyā Ser. A 54 252-259. · Zbl 0761.62040
[13] Dabrowska, D.M. (1992). Variable bandwidth conditional Kaplan-Meier estimate. Scand. J. Stat. 19 351-361. · Zbl 0768.62024
[14] Dette, H., Neumeyer, N. and Pilz, K.F. (2005). A note on nonparametric estimation of the effective dose in quantal bioassay. J. Amer. Statist. Assoc. 100 503-510. · Zbl 1117.62317 · doi:10.1198/016214504000001493
[15] Dette, H., Neumeyer, N. and Pilz, K.F. (2006). A simple nonparametric estimator of a strictly monotone regression function. Bernoulli 12 469-490. · Zbl 1100.62045 · doi:10.3150/bj/1151525131
[16] Dette, H. and Volgushev, S. (2008). Non-crossing non-parametric estimates of quantile curves. J. R. Stat. Soc. Ser. B Stat. Methodol. 70 609-627. · Zbl 05563361 · doi:10.1111/j.1467-9868.2008.00651.x
[17] Efron, B. (1967). The two-sample problem with censored data. In Proceedings of the Fifth Berkeley Symposium in Mathematical Statistics IV (L. Le Cam and J. Neyman, eds.) 831-853. New York: Prentice Hall.
[18] El Ghouch, A. and Van Keilegom, I. (2008). Non-parametric regression with dependent censored data. Scand. J. Stat. 35 228-247. · Zbl 1157.62022 · doi:10.1111/j.1467-9469.2007.00586.x
[19] Fitzenberger, B. (1997). A guide to censored quantile regressions. In Robust Inference. Handbook of Statist. 15 405-437. Amsterdam: North-Holland. · Zbl 0908.62031 · doi:10.1016/S0169-7161(97)15017-9
[20] Fleming, T.R. and Harrington, D.P. (1991). Counting Processes and Survival Analysis. Wiley Series in Probability and Mathematical Statistics : Applied Probability and Statistics . New York: Wiley. · Zbl 0727.62096
[21] Gill, R. (1983). Large sample behaviour of the product-limit estimator on the whole line. Ann. Statist. 11 49-58. · Zbl 0518.62039 · doi:10.1214/aos/1176346055
[22] Gill, R.D. (1989). Non- and semi-parametric maximum likelihood estimators and the von Mises method. I. Scand. J. Stat. 16 97-128. · Zbl 0688.62026
[23] Gill, R.D. and Johansen, S. (1990). A survey of product-integration with a view toward application in survival analysis. Ann. Statist. 18 1501-1555. · Zbl 0718.60087 · doi:10.1214/aos/1176347865
[24] Gu, M.G. and Zhang, C.H. (1993). Asymptotic properties of self-consistent estimators based on doubly censored data. Ann. Statist. 21 611-624. · Zbl 0788.62029 · doi:10.1214/aos/1176349140
[25] Hardy, G.H., Littlewood, J.E. and Pólya, G. (1988). Inequalities. Cambridge Mathematical Library . Cambridge: Cambridge Univ. Press. Reprint of the 1952 edition. · Zbl 0634.26008
[26] Koenker, R. (2005). Quantile Regression. Econometric Society Monographs 38 . Cambridge: Cambridge Univ. Press. · Zbl 1111.62037
[27] Koenker, R. (2008). Censored quantile regression redux. Journal of Statistical Software 27 1-25.
[28] Koenker, R. and Bassett, G. Jr. (1978). Regression quantiles. Econometrica 46 33-50. · Zbl 0373.62038 · doi:10.2307/1913643
[29] Koenker, R. and Geling, O. (2001). Reappraising medfly longevity: A quantile regression survival analysis. J. Amer. Statist. Assoc. 96 458-468. · Zbl 1019.62100 · doi:10.1198/016214501753168172
[30] Li, G. and Datta, S. (2001). A bootstrap approach to nonparametric regression for right censored data. Ann. Inst. Statist. Math. 53 708-729. · Zbl 1003.62036 · doi:10.1023/A:1014644700806
[31] Li, G. and Doss, H. (1995). An approach to nonparametric regression for life history data using local linear fitting. Ann. Statist. 23 787-823. · Zbl 0852.62037 · doi:10.1214/aos/1176324623
[32] Lorentz, G.G. (1953). An inequality for rearrangements. Amer. Math. Monthly 60 176-179. · Zbl 0050.28201 · doi:10.2307/2307574
[33] McKeague, I.W. and Utikal, K.J. (1990). Inference for a nonlinear counting process regression model. Ann. Statist. 18 1172-1187. · Zbl 0721.62087 · doi:10.1214/aos/1176347745
[34] Meeker, W.Q. and Escobar, L.A. (1998). Statistical Methods for Reliability Data. Wiley Series in Probability and Mathematical Statistics , Applied Section xxii 680 p. New York: Wiley. · Zbl 0949.62086
[35] Patilea, V. and Rolin, J.M. (2001). Product-limit estimators of the survival function for doubly censored data. DP 0131, Institut de Statistique, Louvain-la-Neuve.
[36] Patilea, V. and Rolin, J.M. (2006). Product-limit estimators of the survival function with twice censored data. Ann. Statist. 34 925-938. · Zbl 1092.62099 · doi:10.1214/009053606000000065
[37] Peng, L. and Huang, Y. (2008). Survival analysis with quantile regression models. J. Amer. Statist. Assoc. 103 637-649. · Zbl 1408.62159 · doi:10.1198/016214508000000355
[38] Pollard, D. (1984). Convergence of Stochastic Processes. Springer Series in Statistics . New York: Springer. · Zbl 0544.60045
[39] Portnoy, S. (2003). Censored regression quantiles. J. Amer. Statist. Assoc. 98 1001-1012. · Zbl 1045.62099 · doi:10.1198/016214503000000954
[40] Powell, J.L. (1984). Least absolute deviations estimation for the censored regression model. J. Econometrics 25 303-325. · Zbl 0571.62100 · doi:10.1016/0304-4076(84)90004-6
[41] Powell, J.L. (1986). Censored regression quantiles. J. Econometrics 32 143-155. · Zbl 0605.62139 · doi:10.1016/0304-4076(86)90016-3
[42] Shorack, G.R. and Wellner, J.A. (1986). Empirical Processes with Applications to Statistics. Wiley Series in Probability and Mathematical Statistics : Probability and Mathematical Statistics . New York: Wiley. · Zbl 1170.62365
[43] Turnbull, B.W. (1974). Nonparametric estimation of a survivorship function with doubly censored data. J. Amer. Statist. Assoc. 69 169-173. · Zbl 0281.62044 · doi:10.2307/2285518
[44] van Zuijlen, M.C.A. (1978). Properties of the empirical distribution function for independent nonidentically distributed random variables. Ann. Probab. 6 250-266. · Zbl 0396.60040 · doi:10.1214/aop/1176995571
[45] Volgushev, S. and Dette, H. (2012). Nonparametric quantile regression for twice censored data. Technical report. Available at [stat.ME]. 1007.3376v2 · Zbl 1273.62092
[46] Wang, J.G. (1987). A note on the uniform consistency of the Kaplan-Meier estimator. Ann. Statist. 15 1313-1316. · Zbl 0631.62043 · doi:10.1214/aos/1176350507
[47] Yang, S. (1999). Censored median regression using weighted empirical survival and hazard functions. J. Amer. Statist. Assoc. 94 137-145. · Zbl 0997.62080 · doi:10.2307/2669689
[48] Ying, Z., Jung, S.H. and Wei, L.J. (1995). Survival analysis with median regression models. J. Amer. Statist. Assoc. 90 178-184. · Zbl 0818.62103 · doi:10.2307/2291141
[49] Yu, K. and Jones, M.C. (1997). A comparison of local constant and local linear regression quantile estimators. Comput. Statist. Data Anal. 25 159-166. · Zbl 0900.62182 · doi:10.1016/S0167-9473(97)00006-6
[50] Yu, K. and Jones, M.C. (1998). Local linear quantile regression. J. Amer. Statist. Assoc. 93 228-237. · Zbl 0906.62038 · doi:10.2307/2669619
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