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Simple nonparametric estimators of the bivariate survival function under random left truncation and right censoring

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Abstract

We consider estimating the bivariate survival function when both components are subject to random left truncation and right censoring. Using the idea of Sankran and Antony (Sankhyã 69:425–447, 2007) in the competing risks set up, we propose two types of estimators as generalizations of the Dabrowska (Ann Stat 18:1475–1489, 1988) and Campbell and Földes (Nonparametric statistical inference, North-Holland, Amsterdam 1982) estimators. The proposed estimators are easy to implement and do not require iteration. The consistency of the proposed estimators is established. Simulation results indicate that the proposed estimators can outperform the estimators of Shen and Yan (J Stat Plan Inference 138:4041–4054, 2008), which require complex iteration.

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References

  • Campbell G, Földes A (1982) Large sample properties of nonparametric statistical inference. In: Gnredenko BV, Puri ML, Vineze I (eds) Nonparametric statistical inference. North-Holland, Amsterdam, p 103

    Google Scholar 

  • Dabrowska DM (1988) Kaplan–Meier estimate on the plane. Ann Stat 18:1475–1489

    Article  MathSciNet  Google Scholar 

  • Dabrowska DM (1989) Kaplan–Meier estimate on the plane: weak convergence. LIL, and the bootstrap. J Multivar Anal 29:308–325

    Article  MATH  MathSciNet  Google Scholar 

  • Gijbels I, Gürler Ü (1998) Covariance function of a bivariate distribution function estimator for left truncated and right censored data. Stat Sinica 8:1219–1232

    MATH  Google Scholar 

  • Gijbels I, Gürler Ü (1996) Covariance function of a bivariate distribution function estimator for left truncated and right censored data. Discussion paper NO. 9703, 1996, Institute of Statistics. Catholic University of Louvain, Louvain-la-Neuve, Belgium

  • Gill RD, van der Laan MJ, Wellner JA (1995) Inefficient estimators of the bivariate survival function for three models. Annales de l’Institut Henri Poincare. Section B. Prob et stat 31:547–597

    Google Scholar 

  • Gross ST, Lai TL (1996) Bootstrap methods for truncated data and censored data. Stat Sinica 6:509–530

    MATH  MathSciNet  Google Scholar 

  • Gürler Ü (1996) Bivariate estimation with right truncated data. J Am Stat Assoc 91:1152–1165

    MATH  Google Scholar 

  • Gürler Ü (1997) Bivariate distribution and hazard functions when a component is randomly truncated. J Multivar Anal 60:20–47

    Article  MATH  Google Scholar 

  • van der Laan MJ (1996a) Nonparametric estimation of the bivariate survival function with truncated data. J Multivar Anal 58:107–131

    Article  MATH  Google Scholar 

  • van der Laan MJ (1996b) Efficient estimation in the bivariate censoring model and repairing NPMLE. Ann Stat 24:596–627

    Article  MATH  Google Scholar 

  • van der Laan MJ (1997) Nonparametric estimators of the bivariate survival function under random censoring. Stat Nederlandica 51:178–200

    Article  MATH  Google Scholar 

  • Prentice RL, Cai J (1992) Covariance and survivor function estimation using censored multivariate failure time data. Biometrika 79:495–512

    Article  MATH  MathSciNet  Google Scholar 

  • Pruitt RC (1991) Strong consistency of self-consistent estimators:general theory and an application to bivariate survival analysis. Technical Report nr. 543, University of Minnesota

  • Pruitt RC (1993) Small sample comparisons of six bivariate survival curve estimators. J Stat Comput Simul 45:147–167

    Article  MATH  Google Scholar 

  • Sankran PG, Antony AA (2007) Bivariate competing risks models under random left truncation and right censoring. Sankhyã 69:425–447

    Google Scholar 

  • Shen P-S (2006) An inverse-probability-weighted approach to estimation of the bivariate survival function under left-truncation and right-censoring. J Stat Plan Inference 136:4365–4384

    Article  MATH  Google Scholar 

  • Shen P-S, Yan YF (2008) Nonparametric estimation of the bivariate survival function with left-truncated and right-censored data. J Stat Plan Inference 138:4041–4054

    Article  MATH  MathSciNet  Google Scholar 

  • van der Vaart A, Wellner JA (2000) Weak convergence and Empirical process with application to statistics. Springer, Berlin

    Google Scholar 

  • Wang MC (1991) Nonparametric estimation from cross-sectional survival data. J Am Stat Assoc 86:130–143

    Article  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Statistics, Tunghai University, Taichung , 40704, Taiwan

    Pao-sheng Shen

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  1. Pao-sheng Shen
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Correspondence to Pao-sheng Shen.

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Shen, Ps. Simple nonparametric estimators of the bivariate survival function under random left truncation and right censoring. Comput Stat 29, 641–659 (2014). https://doi.org/10.1007/s00180-013-0455-0

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  • DOI: https://doi.org/10.1007/s00180-013-0455-0

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