Advertisement

Chinese Annals of Mathematics, Series B

, Volume 36, Issue 6, pp 969–990 | Cite as

Conditional quantile estimation with truncated, censored and dependent data

  • Hanying Liang
  • Deli Li
  • Tianxuan Miao
Article
  • 61 Downloads

Abstract

This paper deals with the conditional quantile estimation based on left-truncated and right-censored data. Assuming that the observations with multivariate covariates form a stationary α-mixing sequence, the authors derive the strong convergence with rate, strong representation as well as asymptotic normality of the conditional quantile estimator. Also, a Berry-Esseen-type bound for the estimator is established. In addition, the finite sample behavior of the estimator is investigated via simulations.

Keywords

Berry-Esseen-type bound Conditional quantile estimator Strong representation Truncated and censored data α-mixing 

2000 MR Subject Classification

62N02 62G20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Akritas, M. G. and LaValley, M. P., A generalized product-limit estimator for truncated data, J. Nonparametric Statist., 17, 2005, 643–663.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    Beran, R., Nonparametric regression with randomly censored survival data, Technical Report, Department of Statistics, University of California, Berkeley, 1981.Google Scholar
  3. [3]
    Cai, T., Wei, L. J. and Wilcox, M., Semiparametric regression analysis for clustered survival data, Biometrika, 87, 2000, 867–878.zbMATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    Cai, Z. W., Regression quantiles for time series, Econometric Theory, 18, 2002, 169–192.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    Cai, J. and Kim, J., Nonparametric quantile estimation with correlated failure time data, Lifetime Data Analysis, 9, 2003, 357–371.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    Dabrowska, D., Nonparametric quantile regression with censored data, Sankhyā, Ser. A, 54, 1992, 252–259.zbMATHMathSciNetGoogle Scholar
  7. [7]
    Doukhan, P., Mixing: Properties and Examples, Lecture Notes in Statistics, Vol. 85, Springer-Verlag, Berlin, 1994.CrossRefGoogle Scholar
  8. [8]
    Ferraty, F., Rabhi, A. and Vieu, P., Conditional quantiles for dependent functional data with application to the climatic El Ni˜no phenomenon, Sankhyā, 67, 2005, 378–398.zbMATHMathSciNetGoogle Scholar
  9. [9]
    González-Manteiga, W. and Cadarso-Suárez, Z., Asymptotic properties of a generalized Kaplan-Meier estimator with some applications, J. Nonparametric Statist., 4, 1994, 65–78.zbMATHCrossRefGoogle Scholar
  10. [10]
    Gürler, Ü., Stute, W. and Wang, J. L., Weak and strong quantile representations for randomly truncated data with applications, Statist. Probab. Lett., 17, 1993, 139–148.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    Hall, P. and Heyde, C. C., Martingale Limit Theory and Its Application, Academic Press, New York, 1980.zbMATHGoogle Scholar
  12. [12]
    Honda, T., Nonparametric estimation of a conditional quantile for a-mixing processes, Ann. Inst. Statist. Math., 52, 2000, 459–470.zbMATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    Iglesias-Pérez, C. and González-Manteiga, W., Strong representation of a generalized product-limit estimator for truncated and censored data with some applications, J. Nonparametric Statist., 10, 1999, 213–244.zbMATHCrossRefGoogle Scholar
  14. [14]
    Iglesias-Pérez, C., Strong representation of a conditional quantile function estimator with truncated and censored data, Statist. Probab. Lett., 65(2), 2003, 79–91.zbMATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    Kang, S. S. and Koehler, K. J., Modification of the Greenwood formula for correlated failure times, Biometrics, 53, 1997, 885–899.zbMATHCrossRefGoogle Scholar
  16. [16]
    Lemdani, M., Ould-Säid, E. and Poulin, N., Asymptotic properties of a conditional quantile estimator with randomly truncated data, J. Multivar. Analysis, 100, 2009, 546–559.zbMATHCrossRefGoogle Scholar
  17. [17]
    Liang, H. Y. and Fan, G. L., Berry-Esseen-type bounds of estimators in a semiparametric model with linear process errors, J. Multivar. Analysis, 100, 2009, 1–15.zbMATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    Liang, H. Y. and de U˜na-Álvarez, J., Asymptotic properties of conditional quantile estimator for censored dependent observations, Ann. Inst. Statist. Math., 63, 2011, 267–289.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Liang, H. Y. and de U˜na-Álvarez, J., Conditional quantile estimation with auxiliary information for lefttruncated and dependent data, J. Statist. Plan. Inference, 141, 2011, 3475–3488.zbMATHCrossRefGoogle Scholar
  20. [20]
    Liang, H. Y., de U˜na-Álvarez, J. and Iglesias-Pérez, C., Asymptotic properties of conditional distribution estimator with truncated, censored and dependent data, Test, 21(4), 2012, 790–810.zbMATHMathSciNetCrossRefGoogle Scholar
  21. [21]
    Lo, S. and Singh, K., The product-limit estimator and the bootstrap: Some asymptotic representations, Probab. Theory Related Fields, 71, 1985, 455–465.MathSciNetCrossRefGoogle Scholar
  22. [22]
    Michel, R. and Pfanzagl, J., The accuracy of the normal approximation for minimum constrast estimates, Z. Wahrsch. Verw. Gebiete, 18, 1971, 73–84.MathSciNetCrossRefGoogle Scholar
  23. [23]
    Ould-Säid, E., A strong uniform convergence rate of kernel conditional quantile estimator under random censorship, Statist. Probab. Lett., 76, 2006, 579–586.MathSciNetCrossRefGoogle Scholar
  24. [24]
    Ould-Säid, E., Yahia, D. and Necir, A., A strong uniform convergence rate of a kernel conditional quantile estimator under random left-truncation and dependent data, Electronic J. Statist., 3, 2009, 426–445.CrossRefGoogle Scholar
  25. [25]
    Petrov, V. V., Limit Theorems of Probability Theory, Oxford Univ. Press Inc., New York, 1995.zbMATHGoogle Scholar
  26. [26]
    Polonik, W. and Yao, Q., Set-indexed conditional empirical and quantile processes based on dependent data, J. Multivar. Analysis, 80, 2002, 234–255.zbMATHMathSciNetCrossRefGoogle Scholar
  27. [27]
    Shao, Q. and Yu, H., Weak convergence for weighted empirical processes of dependent sequences, Ann. Probab., 24, 1996, 2098–2127.zbMATHMathSciNetCrossRefGoogle Scholar
  28. [28]
    Van Keilegom, I. and Veraverbeke, N., Bootstrapping quantiles in a fixed design regression model with censored data, J. Statist. Plan. Inference, 69, 1998, 115–131.zbMATHCrossRefGoogle Scholar
  29. [29]
    Withers, C. S., Conditions for linear processes to be strong mixing, Z. Wahrsch. Verw. Gebiete, 57, 1981, 477–480.zbMATHMathSciNetCrossRefGoogle Scholar
  30. [30]
    Yang, S. C. and Li, Y. M., Uniformly asymptotic normality of the regression weighted estimator for strong mixing samples, Acta Math. Sinica, 49(5), 2006, 1163–1170.zbMATHMathSciNetGoogle Scholar
  31. [31]
    Zhou, X., Sun, L. Q. and Ren, H., Quantile estimation for left truncted and right censored data, Statist. Sinica, 10, 2000, 1217–1229.zbMATHMathSciNetGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsTongji UniversityShanghaiChina
  2. 2.Department of Mathematical SciencesLakehead UniversityThunder Bay, OntarioCanada

Personalised recommendations