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A Strong Representation of the Product-Limit Estimator for Left Truncated and Right Censored Data

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  • Zhou, Yong
  • Yip, Paul S. F.

Abstract

In this paper we consider the TJW product-limit estimatorFn(x) of an unknown distribution functionFwhen the data are subject to random left truncation and right censorship. An almost sure representation of PL-estimatorFn(x) is derived with an improved error bound under some weaker assumptions. We obtain the strong approximation ofFn(x)-F(x) by Gaussian processes and the functional law of the iterated logarithm is proved for maximal derivation of the product-limit estimator toF. A sharp rate of convergence theorem concerning the smoothed TJW product-limit estimator is obtained. Asymptotic properties of kernel estimators of density function based on TJW product-limit estimator is given.

Suggested Citation

  • Zhou, Yong & Yip, Paul S. F., 1999. "A Strong Representation of the Product-Limit Estimator for Left Truncated and Right Censored Data," Journal of Multivariate Analysis, Elsevier, vol. 69(2), pages 261-280, May.
    • Handle: RePEc:eee:jmvana:v:69:y:1999:i:2:p:261-280
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    References listed on IDEAS

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    1. Arcones, Miguel A. & Giné, Evarist, 1995. "On the law of the iterated logarithm for canonical U-statistics and processes," Stochastic Processes and their Applications, Elsevier, vol. 58(2), pages 217-245, August.
    2. Zhou, Yong, 1996. "A note on the TJW product-limit estimator for truncated and censored data," Statistics & Probability Letters, Elsevier, vol. 26(4), pages 381-387, March.
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    5. Einmahl, J.H.J. & Deheuvels, P, 1996. "On the strong limiting behavior of local functionals of empirical processes based upon censored data," Other publications TiSEM eac4a4cd-81ee-4107-8c70-a, Tilburg University, School of Economics and Management.
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    Cited by:

    1. Ghalibaf, M. Bolbolian & Fakoor, V. & Azarnoosh, H.A., 2010. "Strong Gaussian approximations of product-limit and quantile processes for truncated data under strong mixing," Statistics & Probability Letters, Elsevier, vol. 80(7-8), pages 581-586, April.
    2. Jacobo Uña-álvarez, 2004. "Nonparametric estimation under length-biased sampling and Type I censoring: A moment based approach," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(4), pages 667-681, December.
    3. Goele Massonnet & Paul Janssen & Tomasz Burzykowski, 2008. "Fitting Conditional Survival Models to Meta‐Analytic Data by Using a Transformation Toward Mixed‐Effects Models," Biometrics, The International Biometric Society, vol. 64(3), pages 834-842, September.
    4. Carla Moreira & Jacobo Uña-Álvarez & Ingrid Keilegom, 2014. "Goodness-of-fit tests for a semiparametric model under random double truncation," Computational Statistics, Springer, vol. 29(5), pages 1365-1379, October.
    5. Moreira, Carla & de Una-Alvarez, Jacobo & Van Keilegom, Ingrid, 2012. "Goodness-of-fit Tests for a Semiparametric Model under Random Double Truncation," LIDAM Discussion Papers ISBA 2012024, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. 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.
    7. 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.
    8. Chengbo Li & Yong Zhou, 2021. "The estimation for the general additive–multiplicative hazard model using the length-biased survival data," Statistical Papers, Springer, vol. 62(1), pages 53-74, February.
    9. 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.
    10. Carla Moreira & Jacobo de Uña-Álvarez, 2010. "Bootstrapping the NPMLE for doubly truncated data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(5), pages 567-583.
    11. Zhao, Mu & Bai, Fangfang & Zhou, Yong, 2011. "Relative deficiency of quantile estimators for left truncated and right censored data," Statistics & Probability Letters, Elsevier, vol. 81(11), pages 1725-1732, November.
    12. Sun, Liuquan, 2006. "The strong law under a semiparametric model for truncated and censored data," Statistics & Probability Letters, Elsevier, vol. 76(14), pages 1550-1558, August.
    13. Xun, Li & Shao, Li & Zhou, Yong, 2017. "Efficiency of estimators for quantile differences with left truncated and right censored data," Statistics & Probability Letters, Elsevier, vol. 121(C), pages 29-36.
    14. Shi, Jianhua & Ma, Huijuan & Zhou, Yong, 2018. "The nonparametric quantile estimation for length-biased and right-censored data," Statistics & Probability Letters, Elsevier, vol. 134(C), pages 150-158.
    15. Gneyou, Kossi Essona, 2014. "A strong linear representation for the maximum conditional hazard rate estimator in survival analysis," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 10-18.
    16. Jacobo Uña-Álvarez, 2002. "Product-limit estimation for length-biased censored data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 11(1), pages 109-125, June.
    17. Jacobo de Uña‐Álvarez & Micha Mandel, 2018. "Nonparametric estimation of transition probabilities for a general progressive multi‐state model under cross‐sectional sampling," Biometrics, The International Biometric Society, vol. 74(4), pages 1203-1212, December.

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