Probability Theory and Related Fields

, Volume 71, Issue 3, pp 455–465 | Cite as

The product-limit estimator and the bootstrap: Some asymptotic representations

  • Shaw-Hwa Lo
  • Kesar Singh


The product-limit estimator and its quantile process are represented as i.i.d. mean processes, with a remainder of ordern−3/4(logn)3/4 a.s. Corresponding bootstrap versions of these representations are given, which can help one visualize how the bootstrap procedure operates in this set up.


Stochastic Process Probability Theory Mathematical Biology Asymptotic Representation Bootstrap Procedure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Aly, E.E.A.A., Csörgő, M., Horváth, L.: Strong approximation of the quantile process of the product-limit estimator. Preprint (1983)Google Scholar
  2. Bickel, P.J., Freedman, D.: Some asymptotic theory for the bootstrap. Ann. Statist.9, 1196–1217 (1981)Google Scholar
  3. Billingsley, P.: Convergence of Probability Measures. New York: Wiley 1968Google Scholar
  4. Breslow, N., Crowley, J.: A large sample study of the life table and product-limit estimates under random censorship. Ann. Statist.2, 437–443 (1974)Google Scholar
  5. Burke, M.D., Csögő, S., Horváth, L.: Strong approximations of some biometric estimates under random censorship. Z. Wahrscheinlichkeitstheor. und Verw. Gebiete56, 87–112 (1981)Google Scholar
  6. Cheng, K.F.: On almost sure representation for quantiles of the product limit estimator with application. SANKHYA, A, Vol.46, part 3, 426–443 (1984)Google Scholar
  7. Csörgő, S., Horváth, L.: The Rate of Strong Uniform Consistence for the Product-Limit Estimator. Z. Wahrscheinlichkeitstheor. und Verw. Gebiete62, 411–426 (1983)Google Scholar
  8. Efron, B.: Censored data and the bootstrap. J. Am. Stat. Assoc. 312–321 (1981)Google Scholar
  9. Földes, A., Rejtő, L.: A LIL type result for the product limit estimator. Z. Wahrscheinlichkeitstheorie verw. Gebiete56, 75–86 (1981)Google Scholar
  10. Gill, R.D.: Censoring and stochastic integrals. Mathematical centre tracts 124, Amsterdam (1980)Google Scholar
  11. Gill, R.D.: Large sample behavior of the product-limit estimator on the whole line. Ann. Statist.,11, 49–58 (1983)Google Scholar
  12. Kaplan, E.L., Meier, P.: Nonparametric estimation from incomplete observations. J. Am. Stat. Assoc.53, 457–481 (1958)Google Scholar
  13. Lo, S.-H., Phadia, E.: Asymptotically optimal properties of the PL estimator on the reliability theory based on censored data. Technical Report (1984)Google Scholar
  14. Lo, S.-H.: Some representations of the PL estimator under variable censoring. Technical Report (1985)Google Scholar
  15. Peterson Jr., A.V.: Expressing the Kaplan-Meier estimator as a function of empirical subsurvival function. J. Am. Stat. Assoc.72, 854–858 (1977)Google Scholar
  16. Philips, W.: A functional law of the iterated logarithm for empirical distribution functions of weakly dependent random variables. Ann. Probab.5, 319–350 (1977)Google Scholar
  17. Reid, N.: Estimating the median survival time. Biometrica,68, 601–608 (1981)Google Scholar
  18. Sander, J.M.: The weak convergence of quantiles of the product limit estimator. Technical Report No. 5 Dept. of Statistics, Standford University (1975)Google Scholar
  19. Singh, K.: On asymptotic accuracy of efron's bootstrap. Ann. Stat.9, 1187–1195 (1981)Google Scholar
  20. Wang, J.L.: Estimation of an IFRA (or DFRA) distribution based on censored data. Technical Report (1985)Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Shaw-Hwa Lo
    • 1
  • Kesar Singh
    • 1
  1. 1.Department of StatisticsRutgers University, Hill CenterNew BrunswickUSA

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