IDEAS home Printed from https://ideas.repec.org/a/spr/aistmt/v70y2018i4d10.1007_s10463-017-0617-x.html
   My bibliography  Save this article

Hazard rate estimation for left truncated and right censored data

Author

Listed:
  • Sam Efromovich

    (The University of Texas at Dallas)

  • Jufen Chu

    (Novartis Oncology)

Abstract

Left truncation and right censoring (LTRC) presents a unique challenge for nonparametric estimation of the hazard rate of a continuous lifetime because consistent estimation over the support of the lifetime is impossible. To understand the problem and make practical recommendations, the paper explores how the LTRC affects a minimal (called sharp) constant of a minimax MISE convergence over a fixed interval. The corresponding theory of sharp minimax estimation of the hazard rate is presented, and it shows how right censoring, left truncation and interval of estimation affect the MISE. Obtained results are also new for classical cases of censoring or truncation and some even for the case of direct observations of the lifetime of interest. The theory allows us to propose a relatively simple data-driven estimator for small samples as well as the methodology of choosing an interval of estimation. The estimation methodology is tested numerically and on real data.

Suggested Citation

  • Sam Efromovich & Jufen Chu, 2018. "Hazard rate estimation for left truncated and right censored data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(4), pages 889-917, August.
    • Handle: RePEc:spr:aistmt:v:70:y:2018:i:4:d:10.1007_s10463-017-0617-x
    DOI: 10.1007/s10463-017-0617-x
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10463-017-0617-x
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10463-017-0617-x?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Efromovich S., 2001. "Density Estimation Under Random Censorship and Order Restrictions: From Asymptotic to Small Samples," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 667-684, June.
    2. Shi, Jianhua & Chen, Xiaoping & Zhou, Yong, 2015. "The strong representation for the nonparametric estimator of length-biased and right-censored data," Statistics & Probability Letters, Elsevier, vol. 104(C), pages 49-57.
    3. Yuanjia Wang & Baosheng Liang & Xingwei Tong & Karen Marder & Susan Bressman & Avi Orr-Urtreger & Nir Giladi & Donglin Zeng, 2015. "Efficient estimation of nonparametric genetic risk function with censored data," Biometrika, Biometrika Trust, vol. 102(3), pages 515-532.
    4. Ricardo Cao & Paul Janssen & Noël Veraverbeke, 2005. "Relative hazard rate estimation for right censored and left truncated data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 14(1), pages 257-280, June.
    5. Talamakrouni, Majda & Van Keilegom, Ingrid & El Ghouch, Anouar, 2016. "Parametrically guided nonparametric density and hazard estimation with censored data," Computational Statistics & Data Analysis, Elsevier, vol. 93(C), pages 308-323.
    6. A. Antoniadis & G. Grégoire & G. Nason, 1999. "Density and hazard rate estimation for right‐censored data by using wavelet methods," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(1), pages 63-84.
    7. Chiung-Yu Huang & Jing Qin, 2013. "Semiparametric estimation for the additive hazards model with left-truncated and right-censored data," Biometrika, Biometrika Trust, vol. 100(4), pages 877-888.
    8. Bremhorst, Vincent & Lambert, Philippe, 2016. "Flexible estimation in cure survival models using Bayesian P-splines," Computational Statistics & Data Analysis, Elsevier, vol. 93(C), pages 270-284.
    9. XiaoFei, Lu & Min, Liu, 2014. "Hazard rate function in dynamic environment," Reliability Engineering and System Safety, Elsevier, vol. 130(C), pages 50-60.
    10. Talamakrouni, Majda & Van Keilegom, Ingrid & El Ghouch, Anouar, 2016. "Parametrically guided nonparametric density and hazard estimation with censored data," LIDAM Reprints ISBA 2016001, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    11. Qian, Jing & Betensky, Rebecca A., 2014. "Assumptions regarding right censoring in the presence of left truncation," Statistics & Probability Letters, Elsevier, vol. 87(C), pages 12-17.
    12. Sam Efromovich, 2016. "Minimax theory of nonparametric hazard rate estimation: efficiency and adaptation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(1), pages 25-75, February.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Moawia Alghalith, 2022. "Methods in Econophysics: Estimating the Probability Density and Volatility," Papers 2301.10178, arXiv.org.
    2. Yu Shen & Jing Ning & Jing Qin, 2017. "Nonparametric and semiparametric regression estimation for length-biased survival data," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 23(1), pages 3-24, January.
    3. Narisetty, Naveen & Koenker, Roger, 2022. "Censored quantile regression survival models with a cure proportion," Journal of Econometrics, Elsevier, vol. 226(1), pages 192-203.
    4. Coria, V.H. & Maximov, S. & Rivas-Dávalos, F. & Melchor, C.L. & Guardado, J.L., 2015. "Analytical method for optimization of maintenance policy based on available system failure data," Reliability Engineering and System Safety, Elsevier, vol. 135(C), pages 55-63.
    5. Reder, Maik & Yürüşen, Nurseda Y. & Melero, Julio J., 2018. "Data-driven learning framework for associating weather conditions and wind turbine failures," Reliability Engineering and System Safety, Elsevier, vol. 169(C), pages 554-569.
    6. Ma, Huijuan & Zhang, Feipeng & Zhou, Yong, 2015. "Composite estimating equation approach for additive risk model with length-biased and right-censored data," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 45-53.
    7. Dirick, Lore & Claeskens, Gerda & Vasnev, Andrey & Baesens, Bart, 2022. "A hierarchical mixture cure model with unobserved heterogeneity for credit risk," Econometrics and Statistics, Elsevier, vol. 22(C), pages 39-55.
    8. Chyong-Mei Chen & Pao-sheng Shen & Yi Liu, 2021. "On semiparametric transformation model with LTRC data: pseudo likelihood approach," Statistical Papers, Springer, vol. 62(1), pages 3-30, February.
    9. Michaela Kreyenfeld & Dirk Konietzka & Philippe Lambert & Vincent Jerald Ramos, 2023. "Second Birth Fertility in Germany: Social Class, Gender, and the Role of Economic Uncertainty," European Journal of Population, Springer;European Association for Population Studies, vol. 39(1), pages 1-27, December.
    10. Elisa–María Molanes-López & Ricardo Cao, 2008. "Relative density estimation for left truncated and right censored data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(8), pages 693-720.
    11. Mohamed Elamin Abdallah Mohamed Elamin Omer & Mohd Rizam Abu Bakar & Mohd Bakri Adam & Mohd Shafie Mustafa, 2020. "Cure Models with Exponentiated Weibull Exponential Distribution for the Analysis of Melanoma Patients," Mathematics, MDPI, vol. 8(11), pages 1-15, November.
    12. Prabhashi W. Withana Gamage & Christopher S. McMahan & Lianming Wang, 2023. "A flexible parametric approach for analyzing arbitrarily censored data that are potentially subject to left truncation under the proportional hazards model," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 29(1), pages 188-212, January.
    13. Gressani, Oswaldo & Lambert, Philippe, 2016. "Fast Bayesian inference in semi-parametric P-spline cure survival models using Laplace approximations," LIDAM Discussion Papers ISBA 2016041, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    14. E. Brunel & F. Comte & A. Guilloux, 2009. "Nonparametric density estimation in presence of bias and censoring," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 18(1), pages 166-194, May.
    15. Sam Efromovich, 2007. "Optimal nonparametric estimation of the density of regression errors with finite support," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(4), pages 617-654, December.
    16. Zhiyuan Zuo & Liang Wang & Yuhlong Lio, 2022. "Reliability Estimation for Dependent Left-Truncated and Right-Censored Competing Risks Data with Illustrations," Energies, MDPI, vol. 16(1), pages 1-25, December.
    17. Bella Vakulenko‐Lagun & Jing Qian & Sy Han Chiou & Nancy Wang & Rebecca A. Betensky, 2022. "Nonparametric estimation of the survival distribution under covariate‐induced dependent truncation," Biometrics, The International Biometric Society, vol. 78(4), pages 1390-1401, December.
    18. Sam Efromovich, 2016. "Minimax theory of nonparametric hazard rate estimation: efficiency and adaptation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(1), pages 25-75, February.
    19. Peng Liu & Kwun Chuen Gary Chan & Ying Qing Chen, 2023. "On a simple estimation of the proportional odds model under right truncation," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 29(3), pages 537-554, July.
    20. Bremhorst, Vincent & Kreyenfeld, Michaela & Lambert, Philippe, 2017. "Nonparametric double additive cure survival models: an application to the estimation of the nonlinear effect of age at first parenthood on fertility progression," LIDAM Discussion Papers ISBA 2017004, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:aistmt:v:70:y:2018:i:4:d:10.1007_s10463-017-0617-x. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.