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The strong representation for the nonparametric estimator of length-biased and right-censored data

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  • Shi, Jianhua
  • Chen, Xiaoping
  • Zhou, Yong

Abstract

In this paper, we consider the modified product-limit estimator of an unknown distribution function proposed by Huang and Qin (2011), where the observations are subject to length-biased and right-censored data. A strong representation for the modified product-limit estimator is established with a remainder O(n−3/4(logn)3/4) a.s. Such results are very useful when we consider statistics that are the function of the estimator of nonparametric distribution function. Also, a uniform consistency rate of the estimator is given.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:104:y:2015:i:c:p:49-57
    DOI: 10.1016/j.spl.2015.04.029
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    References listed on IDEAS

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    1. César Sánchez Sellero & Wenceslao González Manteiga & Ingrid Van Keilegom, 2005. "Uniform Representation of Product‐Limit Integrals with Applications," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(4), pages 563-581, December.
    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.
    3. Gijbels, I. & Wang, J. L., 1993. "Strong Representations of the Survival Function Estimator for Truncated and Censored Data with Applications," Journal of Multivariate Analysis, Elsevier, vol. 47(2), pages 210-229, November.
    4. Chiung-Yu Huang & Jing Qin, 2011. "Nonparametric estimation for length-biased and right-censored data," Biometrika, Biometrika Trust, vol. 98(1), pages 177-186.
    5. Shen, Yu & Ning, Jing & Qin, Jing, 2009. "Analyzing Length-Biased Data With Semiparametric Transformation and Accelerated Failure Time Models," Journal of the American Statistical Association, American Statistical Association, vol. 104(487), pages 1192-1202.
    6. Liang, Han-Ying & Li, Deli & Qi, Yongcheng, 2009. "Strong convergence in nonparametric regression with truncated dependent data," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 162-174, January.
    7. 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.
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    Cited by:

    1. 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.
    2. 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.

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