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Nonparametric Estimation of the Bivariate Survival Function with Truncated Data


  • van der Laan, Mark J.


Randomly left or right truncated observations occur when one is concerned with estimation of the distribution of time between two events and when one only observes the time if one of the two events falls in a fixed time-window, so that longer survivial times have higher probability to be part of the sample than short survival times. In important AIDS-applications the time between seroconversion and AIDS is only observed if the person did not die before the start of the time-window. Hence, here the time of interest is truncated if another related time-variable is truncated. This problem is a special case of estimation of the bivariate survival function based on truncation by a bivariate truncation time, the problem covered in this paper; in the AIDS-application one component of the bivariate truncation time- vector is alway zero. In this application the bivariate survival function is of interest itself in order to study the relation between time till AIDS and time between AIDS and death. We provide a quick algorithm for computation of the NPMLE. In particular, it is shown that the NPMLE is explicit for the special case when one of the truncation times is zero, as in the aids-application above. We prove that the NPMLE is consistent under the minimal condition that [integral operator]Â dF/G

Suggested Citation

  • van der Laan, Mark J., 1996. "Nonparametric Estimation of the Bivariate Survival Function with Truncated Data," Journal of Multivariate Analysis, Elsevier, vol. 58(1), pages 107-131, July.
  • Handle: RePEc:eee:jmvana:v:58:y:1996:i:1:p:107-131

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    Cited by:

    1. Larsen Klaus & Petersen Janne & Bernstein Inge & Nilbert Mef, 2009. "A Parametric Model for Analyzing Anticipation in Genetically Predisposed Families," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 8(1), pages 1-11, June.
    2. Shen, Pao-sheng, 2009. "An inverse-probability-weighted approach to the estimation of distribution function with doubly censored data," Statistics & Probability Letters, Elsevier, vol. 79(9), pages 1269-1276, May.
    3. Bella Vakulenko-Lagun & Micha Mandel & Yair Goldberg, 2017. "Nonparametric estimation in the illness-death model using prevalent data," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 23(1), pages 25-56, January.
    4. Pao-sheng Shen, 2010. "Nonparametric analysis of doubly truncated data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(5), pages 835-853, October.
    5. Shen, Pao-sheng, 2010. "Semiparametric estimation of survival function when data are subject to dependent censoring and left truncation," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 161-168, February.
    6. Gürler, Ülkü & Prewitt, Kathryn, 2000. "Bivariate Density Estimation with Randomly Truncated Data," Journal of Multivariate Analysis, Elsevier, vol. 74(1), pages 88-115, July.


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