Advertisement

Asymptotic Theory in Rank Estimation for AFT Model Under Fixed Censorship

  • Zhezhen Jin
  • Zhiliang Ying
Part of the Statistics for Industry and Technology book series (SIT)

Abstract

Rank based parameter estimation in the accelerated failure time model with right-censoring has been studied rigorously by many authors [(1990), (1991) and (1993)]. A key assumption in establishing the asymptotic normality for the rank estimator is that, conditional on covariates, the censoring time has a density function. This assumption may be violated in many situations. It may also be non-intrinsic in that the limiting variance formula does not involve such a density. We show in this paper that the usual asymptotic theory for rank estimation can indeed be established without this stringent and unnatural assumption.

Keywords

Accelerated failure time model asymptotic normality exponential inequality fixed censoring linear rank statistics survival data 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aalen, O. O. (1975). Statistical inference for a family of counting processes, Ph.D. thesis, University of California, Berkeley.Google Scholar
  2. 2.
    Andersen, P. K., Borgan, O., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes, Springer-Verlag, New York.zbMATHCrossRefGoogle Scholar
  3. 3.
    Cox, D. R. and Oakes, D. (1984). Analysis of Survival Data, Chapman & Hall, London, England.Google Scholar
  4. 4.
    Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis, John Wiley & Sons, New York.zbMATHGoogle Scholar
  5. 5.
    Gehan, E. A. (1965). A generalized Wilcoxon test for comparing arbitrarily single-censored samples, Biometrika, 52, 203–223.zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Gross, S. T. and Huber-Carol, C. (1992). Regression models for truncated survival data, Scandinavian Journal of Statistics, 19, 193–213.zbMATHMathSciNetGoogle Scholar
  7. 7.
    Gross, S. T. and Lai, T. L. (1996). Nonparametric estimation and regression analysis with left-truncated and right-censored data, Journal of the American Statistical Association, 91, 1166–1180.zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Huber, C. (2000). Censored and truncated lifetime data, In Recent Advances in Reliability Theory (Eds., N. Limnios and M. Nikulin), pp. 291–305, Birkhäuser, Boston, Massachusetts.CrossRefGoogle Scholar
  9. 9.
    Jin, Z., Lin, D.Y., Wei, L. J. and Ying, Z. (2003). Rank-based inference for the accelerated failure time models, Biometrika, 90, 341–353.zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Kalbfleisch, J. D. and Prentice, R. L. (2002). The Statistical Analysis of Failure Time Data, Second edition, John Wiley & Sons, New York.zbMATHCrossRefGoogle Scholar
  11. 11.
    Klein, J.P. and Moeschberger, M.L. (1997). Survival Analysis. Techniques for censored and truncated data, Springer-Verlag, New York.zbMATHGoogle Scholar
  12. 12.
    Lai, T. L. and Ying, Z. (1991). Rank regression methods for left-truncated and right-censored data, Annals of Statistics, 19, 531–556.zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Lai, T. L. and Ying, Z. (1992). Linear rank statistics in regression analysis with censored or truncated data, Journal of Multivariate Analaysis, 40, 13–45.zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Mantel, N. and Haenszel, W. (1959). Statistical aspects of the analysis of data from retrospective studies of disease, Journal of National Cancer Institute, 22, 719–748.Google Scholar
  15. 15.
    Miller, R. G. (1981). Survival Analysis, John Wiley & Sons, New York.zbMATHGoogle Scholar
  16. 16.
    Nolan, D. and Pollard, D. (1987). U-processes: rates of convergence, Annals of Statistics, 15, 780–799.zbMATHMathSciNetGoogle Scholar
  17. 17.
    Pakes, A. and Pollard, D. (1989). Simulation and the asymptotics of optimization estimators, Econometrica, 57, 1027–2057.zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Peto, R. and Peto, J. (1972). Asymptotically efficient rank invariant test procedures (with discussion), Journal of the Royal Statistical Society, Series A, 135, 185–206.CrossRefGoogle Scholar
  19. 19.
    Pollard, D. (1990). Empirical Processes: Theory and Application. NSFCBMS regional conference series in Probability and Statistics, Vol 2, Institute of Mathematical Statistics and American Statistical Association, Hayward, California.Google Scholar
  20. 20.
    Prentice, R. L. (1978). Linear rank tests with right censored data, Biometrika, 65, 167–79.zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Ritov, Y. (1990). Estimation in a linear regression model with censored data, Annals of Statistics, 18, 303–28.zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Tsiatis, A. A. (1990). Estimating regression parameters using linear rank tests for censored data, Annals of Statistics, 18, 354–72.zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes, Springer-Verlag, New York.zbMATHCrossRefGoogle Scholar
  24. 24.
    Ying, Z. (1993). A large sample study of rank estimation for censored regression data, Annals of Statistics, 21, 76–99.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Zhezhen Jin
    • 1
  • Zhiliang Ying
    • 1
  1. 1.Department of Bio statisticsColumbia UniversityNew YorkUSA

Personalised recommendations