Skip to main content
Log in

Product-limit estimation for length-biased censored data

Test Aims and scope Submit manuscript

Abstract

Length-biased and censored data may appear when analyzing times of duration In this work, a new empirical curve \(\tilde F\) for approximating a distribution functionF under right-censoring and length-bias is introduced. The proposed estimate is (not equal to but) closely related to the product-limit Kaplan-Meier estimator. Strong consistency and distributional convergence is established for a general empirical parameter \(\tilde \gamma = g\left( {\int {\varphi _1 d\tilde F} ,...,\int {\varphi _\tau d\tilde F} } \right)\). As applications, one can obtain the corresponding large sample results for estimates of the distribution function, the cumulative harard function, and the mean residual time function. The new method is illustrated with real data concerning unempoloyment duration.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  • Chaudhuri, P. andMarron, J. S. (1999). SiZer for exploration of structure in curves.Journal of the American Statistical Association, 94:807–823.

    Article  MATH  MathSciNet  Google Scholar 

  • De Uña Álvarez, J. (2001). On efficiency under selection bias caused by truncation.Unpublished manuscript.

  • de Uña Álvarez, J., Otero-Giráldez, M. S., andÁlvarez Llorente, G. (2001). Nonparametric estimation for length-biased rigth-censored data: an application to unemployment duration analysis for married woment.Unpublished manuscript.

  • Horváth, J. (1985). Estimation from a length-biased distribution.Statistics & Decision, 3(1–2):91–113.

    MATH  Google Scholar 

  • Hughes, J. W. andSavoca, E. (1999). Accounting for censoring in duration data: an application to estimating the effect of legal reforms on the duration of medical malpractice disputes.Journal of Applied Statistics, 26:219–228.

    Article  MATH  Google Scholar 

  • Jones, M. C. (1991). Kernel density estimation for lenghh biased data.Biometrika 78:511–519.

    Article  MathSciNet  Google Scholar 

  • Kalbfleisch, J. D. andPrentice, R. L. (1980).The Statistical Analysis of Failure Time Data. Wiley, New York.

    MATH  Google Scholar 

  • Kaplan, E. andMeier, P. (1958). Nonparametric estimation from incomplete obervations.Journal of the American Statistical Association, 53:457–481.

    Article  MATH  MathSciNet  Google Scholar 

  • Lai, T. L. andYing, Z. (1991). Estimating a distribution function with truncated and censored data.Annals of Statistics, 19:417–442.

    MATH  MathSciNet  Google Scholar 

  • Maller, R. andZhou, X. (1996).Survival Analysis with Long-Term Survivors. Wiley, Baffins Lane, Chichester.

    Google Scholar 

  • Marron, J. S. andde Uña Álvarez, J. (2001). SiZer for length biased, censored density and hazard estimation. Reports in Statistics and Operations Research, University of Santiago de Compostela, Spain.

    Google Scholar 

  • Serfling, R. J. (1980).Approximation theorems of mathematical statistics. John Wiley & Sons Inc., New York, 0-471-02403-1. Wiley Series in Probability and Mathematical Statistics.

    MATH  Google Scholar 

  • Simon, R. (1980). Length biased sampling in etiologic studies.American Journal of Epidemiology, 111:444–452.

    Google Scholar 

  • Stute, W. (1995a). The central limit theorem under random censorship.Annals of Statistics, 23(2):422–439.

    MATH  MathSciNet  Google Scholar 

  • Stute, W. (1995b). The statistical analysis of Kaplan-Meier integrals. InAnalysis of censored data (Pune, 1994/1995), pp. 231–254. Inst. Math. Statist., Hayward, CA.

    Google Scholar 

  • Stute, W. (1996). The jackknife estimate of variance of a Kaplan-Meier integral.Annals of Statistics,24(6):2679–2704.

    Article  MATH  MathSciNet  Google Scholar 

  • Stute, W. andWang, J. L. (1993). The strong law under random censorship.Annals of Statistics, 21(3):1591–1607.

    MATH  MathSciNet  Google Scholar 

  • Tsai, W. Y., Jewell, N. P. andWang, M. C. (1987). A note on the product-limit estimator under right censoring and left truncation.Biometrika, 74:883–886.

    Article  MATH  Google Scholar 

  • Vardi, Y. (1982). Nonparametric estimation in the presence of length bias.Annals of Statistics, 10(2):616–620.

    MATH  MathSciNet  Google Scholar 

  • Vardi, Y. (1985). Empirical distributions in selection bias models.Annals of Statistics, 13(1):178–205. With discussion by C. L. Mallows.

    MATH  MathSciNet  Google Scholar 

  • Winter, B. B. andFöldes, A. (1988). A product-limit estimator for use with length-biased data.Canadian Journal of Statistics, 16:337–355.

    MATH  Google Scholar 

  • Woodroofe, M. (1987). Estimating a distribution function with truncated data.Annals of Statistics, 13(1):163–177.

    MathSciNet  Google Scholar 

  • Zhou, Y. (1996). A note on the TJW product-limit estimator for truncated and censored data.Statistics & Probability Letters, 26(4):381–387.

    Article  MATH  MathSciNet  Google Scholar 

  • Zhou, Y. andYip, P. S. F. (1999). A strong representation of the product-limit estimator for left truncated and right censored data.Journal of Multivariate Analysis, 69(2):261–280.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Dedicated to the memory of Alejandro and José-Luis de Uña-Álvarez.

Work supported by the DGES grant PB98-0182-C02-02 and the Xunta de Galicia grant PGIDT00PXI20704PN.

Rights and permissions

Reprints and permissions

About this article

Cite this article

de Uña-Álvarez, J. Product-limit estimation for length-biased censored data. Test 11, 109–125 (2002). https://doi.org/10.1007/BF02595732

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02595732

Key Words

AMS subject classification

Navigation