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Product-limit estimators of the survival function with twice censored data. (English) Zbl 1092.62099
Summary: A model for competing (resp. complementary) risks survival data where the failure time can be left (resp. right) censored is proposed. Product-limit estimators for the survival functions of the individual risks are derived. We deduce the strong convergence of our estimators on the whole real half-line without any additional assumptions and their asymptotic normality under conditions concerning only the observed distribution. When the observations are generated according to the double censoring model introduced by B. W. Turnbull [J. Am. Stat. Assoc. 69, 169–173 (1974; Zbl 0281.62044)], the product-limit estimators represent upper and lower bounds for Turnbull’s estimator.

MSC:
62N01 Censored data models
62G20 Asymptotic properties of nonparametric inference
62N02 Estimation in survival analysis and censored data
62G05 Nonparametric estimation
Software:
SPLIDA
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References:
[1] Basu, A. P. and Ghosh, J. K. (1980). Identifiability of distributions under competing risks and complementary risks model. Comm. Statist. A—Theory Methods 9 1515–1525. · Zbl 0454.62086
[2] Doss, H., Freitag, S. and Prochan, F. (1989). Estimating jointly system and component reliabilities using mutual censorship approach. Ann. Statist. 17 764–782. · Zbl 0693.62080
[3] Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis . Wiley, New York. · Zbl 0727.62096
[4] Gill, R. (1983). Large sample behavior of the product-limit estimator on the whole line. Ann. Statist. 11 49–58. · Zbl 0518.62039
[5] Gill, R. (1989). Non- and semi-parametric maximum likelihood estimators and the von-Mises method. I (with discussion). Scand. J. Statist. 16 97–128. · Zbl 0688.62026
[6] Gill, R. and Johansen, S. (1990). A survey of product-integration with a view toward application in survival analysis. Ann. Statist. 18 1501–1555. · Zbl 0718.60087
[7] Gu, M. G. and Zhang, C.-H. (1993). Asymptotic properties of self-consistent estimators based on doubly censored data. Ann. Statist. 21 611–624. · Zbl 0788.62029
[8] Meeker, W. Q. and Escobar, L. A. (1998). Statistical Methods for Reliability Data. Wiley, New York. · Zbl 0949.62086
[9] Morales, D., Pardo, L. and Quesada, V. (1991). Bayesian survival estimation for incomplete data when the life distribution is proportionally related to the censoring time distribution. Comm. Statist. Theory Methods 20 831–850. · Zbl 0724.62008
[10] Patilea, V. and Rolin, J.-M. (2001). Product-limit estimators of the survival function for doubly censored data. DP 0131, Institut de Statistique, Louvain-la-Neuve. Available at www.stat.ucl.ac.be.
[11] Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York. · Zbl 0544.60045
[12] Rolin, J.-M. (2001). Nonparametric competing risks models: Identification and strong consistency. DP 0115, Institut de Statistique, Louvain-la-Neuve. Available at www.stat.ucl.ac.be.
[13] Stute, W. and Wang, J.-L. (1993). The strong law under random censorship. Ann. Statist. 21 1591–1607. · Zbl 0785.60020
[14] Turnbull, B. W. (1974). Nonparametric estimation of a survivorship function with doubly censored data. J. Amer. Statist. Assoc. 69 169–173. JSTOR: · Zbl 0281.62044
[15] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York. · Zbl 0862.60002
[16] Ying, Z. (1989). A note on the asymptotic properties of the product-limit estimator on the whole line. Statist. Probab. Lett. 7 311–314. · Zbl 0675.62034
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