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Likelihood Ratio Confidence Bands in Non–parametric Regression with Censored Data

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  • GANG LI
  • INGRID VAN KEILEGOM

Abstract

Let (X, Y) be a random vector, where Y denotes the variable of interest possibly subject to random right censoring, and X is a covariate. We construct confidence intervals and bands for the conditional survival and quantile function of Y given X using a non‐parametric likelihood ratio approach. This approach was introduced by Thomas & Grunkemeier (1975), who estimated confidence intervals of survival probabilities based on right censored data. The method is appealing for several reasons: it always produces intervals inside [0, 1], it does not involve variance estimation, and can produce asymmetric intervals. Asymptotic results for the confidence intervals and bands are obtained, as well as simulation results, in which the performance of the likelihood ratio intervals and bands is compared with that of the normal approximation method. We also propose a bandwidth selection procedure based on the bootstrap and apply the technique on a real data set.

Suggested Citation

  • Gang Li & Ingrid Van Keilegom, 2002. "Likelihood Ratio Confidence Bands in Non–parametric Regression with Censored Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 29(3), pages 547-562, September.
    • Handle: RePEc:bla:scjsta:v:29:y:2002:i:3:p:547-562
    DOI: 10.1111/1467-9469.00305
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    Cited by:

    1. Song Chen & Ingrid Van Keilegom, 2009. "A review on empirical likelihood methods for regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 18(3), pages 415-447, November.
    2. Jimin Lee & Seunggeun Hyun, 2011. "Confidence bands for the difference of two survival functions under the additive risk model," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(4), pages 785-797, February.
    3. Yichuan Zhao & Song Yang, 2008. "Empirical likelihood inference for censored median regression with weighted empirical hazard functions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(2), pages 441-457, June.
    4. Ali Al-Sharadqah & Majid Mojirsheibani, 2020. "A simple approach to construct confidence bands for a regression function with incomplete data," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 104(1), pages 81-99, March.
    5. Arteaga-Molina, Luis A. & Rodríguez-Poo, Juan M., 2019. "Empirical likelihood based inference for a categorical varying-coefficient panel data model with fixed effects," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 110-124.
    6. Wang, Qihua & Shen, Junshan, 2008. "Estimation and confidence bands of a conditional survival function with censoring indicators missing at random," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 928-948, May.
    7. Zhao, Yichuan & Zhao, Meng, 2011. "Empirical likelihood for the contrast of two hazard functions with right censoring," Statistics & Probability Letters, Elsevier, vol. 81(3), pages 392-401, March.
    8. Zhao, Yichuan, 2010. "Semiparametric inference for transformation models via empirical likelihood," Journal of Multivariate Analysis, Elsevier, vol. 101(8), pages 1846-1858, September.
    9. Cheng, Guang & Zhao, Yichuan & Li, Bo, 2012. "Empirical likelihood inferences for the semiparametric additive isotonic regression," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 172-182.
    10. Junshan Shen & Shuyuan He, 2008. "Empirical likelihood confidence intervals for hazard and density functions under right censorship," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(3), pages 575-589, September.
    11. Yanqing Sun & Rajeshwari Sundaram & Yichuan Zhao, 2009. "Empirical Likelihood Inference for the Cox Model with Time‐dependent Coefficients via Local Partial Likelihood," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(3), pages 444-462, September.
    12. McKeague, Ian W. & Zhao, Yichuan, 2006. "Width-scaled confidence bands for survival functions," Statistics & Probability Letters, Elsevier, vol. 76(4), pages 327-339, February.
    13. Junshan Shen & Shuyuan He, 2007. "Empirical likelihood for the difference of quantiles under censorship," Statistical Papers, Springer, vol. 48(3), pages 437-457, September.
    14. Zhao, Yichuan & Chen, Feiming, 2008. "Empirical likelihood inference for censored median regression model via nonparametric kernel estimation," Journal of Multivariate Analysis, Elsevier, vol. 99(2), pages 215-231, February.
    15. Shen, Junshan & Yuen, Kam Chuen & Liu, Chunling, 2016. "Empirical likelihood confidence regions for one- or two- samples with doubly censored data," Computational Statistics & Data Analysis, Elsevier, vol. 93(C), pages 285-293.

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