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Kaplan-Meier estimate on the plane: Weak convergence, LIL, and the bootstrap

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  • Dabrowska, Dorota M.

Abstract

We consider estimation of the bivariate survival function F(s,t) under bivariate random right censoring. It is shown that the bivariate product integral estimator can be written as , where is a sum of mean zero iid processes and is a remainder term of order O((n-1logn)1/2 (n-1log logn)1/8) a.s. Using this representation we establish weak convergence of as well as the law of iterated logarithm. Similar results are obtained for the bootstrap version of .

Suggested Citation

  • Dabrowska, Dorota M., 1989. "Kaplan-Meier estimate on the plane: Weak convergence, LIL, and the bootstrap," Journal of Multivariate Analysis, Elsevier, vol. 29(2), pages 308-325, May.
    • Handle: RePEc:eee:jmvana:v:29:y:1989:i:2:p:308-325
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    Cited by:

    1. Chen, Xiaohong & Fan, Yanqin & Pouzo, Demian & Ying, Zhiliang, 2010. "Estimation and model selection of semiparametric multivariate survival functions under general censorship," Journal of Econometrics, Elsevier, vol. 157(1), pages 129-142, July.
    2. Svetlana K. Eden & Chun Li & Bryan E. Shepherd, 2022. "Nonparametric estimation of Spearman's rank correlation with bivariate survival data," Biometrics, The International Biometric Society, vol. 78(2), pages 421-434, June.
    3. Song Zhang & Yang Qu & Yu Cheng & Oscar L. Lopez & Abdus S. Wahed, 2022. "Prognostic accuracy for predicting ordinal competing risk outcomes using ROC surfaces," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 28(1), pages 1-22, January.
    4. Jian-Jian Ren & Tonya Riddlesworth, 2014. "Empirical likelihood bivariate nonparametric maximum likelihood estimator with right censored data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(5), pages 913-930, October.
    5. Emura, Takeshi & Kao, Fan-Hsuan & Michimae, Hirofumi, 2014. "An improved nonparametric estimator of sub-distribution function for bivariate competing risk models," Journal of Multivariate Analysis, Elsevier, vol. 132(C), pages 229-241.
    6. Chen, Xiaohong & Fan, Yanqin, 2007. "A Model Selection Test For Bivariate Failure-Time Data," Econometric Theory, Cambridge University Press, vol. 23(3), pages 414-439, June.
    7. Sanders, Lisanne & Melenberg, Bertrand, 2016. "Estimating the joint survival probabilities of married individuals," Insurance: Mathematics and Economics, Elsevier, vol. 67(C), pages 88-106.
    8. Wang, Qi-Hua, 2000. "Moment and probability inequalities for the bivariate product-limit estimator," Statistics & Probability Letters, Elsevier, vol. 46(1), pages 1-12, January.
    9. Fermanian, Jean-David, 1997. "Multivariate Hazard Rates under Random Censorship," Journal of Multivariate Analysis, Elsevier, vol. 62(2), pages 273-309, August.
    10. Pao-sheng Shen, 2014. "Simple nonparametric estimators of the bivariate survival function under random left truncation and right censoring," Computational Statistics, Springer, vol. 29(3), pages 641-659, June.
    11. Kouros Owzar & Pranab Kumar Sen, 2003. "Copulas: concepts and novel applications," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 323-353.

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