Advertisement

Analysis of Censored Data

  • Mayer Alvo
  • Philip L. H. Yu
Chapter
Part of the Springer Series in the Data Sciences book series (SSDS)

Abstract

Censored data occur when the value of an observation is only partially known. For example, it may be known that someone’s exact wealth is unknown but it may be known that their wealth exceeds one million dollars. In left censoring, the data may fall below a certain value whereas in right censoring, it may be above a certain value. Type I censoring occurs when the subjects of an experiment are right censored. Type II censoring occurs when the experiment stops after a certain number of subjects have failed; the remaining subjects are then right censored. Truncated data occur when observations never lie outside a given range. For example, all data outside the unit interval is discarded. A good example to illustrate the ideas occurs in insurance companies. Left truncation occurs when policyholders are subject to a deductible whereas right censoring occurs when policyholders are subject to an upper pay limit.

References

  1. Aalen, O. (1978). Nonparametric estimation of partial transition probabilities in multiple decrement models. Ann. Statist., 6(3):534–545.MathSciNetCrossRefGoogle Scholar
  2. Alvo, M., Lai, T. L., and Yu, P. L. H. (2018). Parametric embedding of nonparametric inference problems. Journal of Statistical Theory and Practice, 12(1):151–164.MathSciNetCrossRefGoogle Scholar
  3. Andersen, P., Borgan, O., Gill, R., and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer: New York.CrossRefGoogle Scholar
  4. Bhattacharya, P. K., Chernoff, H., and Yang, S. S. (1983). Nonparametric estimation of the slope of a truncated regression. Ann. Statist., 11(2):505–514.MathSciNetCrossRefGoogle Scholar
  5. Bickel, P. J. (1982). On adaptive estimation. Annals of Statistics, 10(3):647–671.MathSciNetCrossRefGoogle Scholar
  6. Bickel, P. J., Klaassen, C. A. J., Ritov, Y., and Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. The Johns Hopkins University Press.Google Scholar
  7. Breslow, N. (1970). A generalized Kruskal-Wallis test for comparing K samples subject to unequal pattern of censorship. Biometrika, 57:579–594.CrossRefGoogle Scholar
  8. Cox, D. R. (1972). Regression models and life-tables. J. Roy. Statist. Soc. Ser. B., 34(2):187–220.MathSciNetzbMATHGoogle Scholar
  9. Cox, D. R. (1975). Partial likelihood. Biometrika, 62(2):269–276.MathSciNetCrossRefGoogle Scholar
  10. Cuzick, J. (1985). Asymptotic properties of censored linear rank tests. Ann. Statist., 13(1):133–141.MathSciNetCrossRefGoogle Scholar
  11. Gehan, E. A. (1965). A generalized Wilcoxon test for comparing arbitrarily singly-censored samples. Biometrika, 52(1/2):203–223.MathSciNetCrossRefGoogle Scholar
  12. Gill, R. D. (1980). Censoring and Stochastic Integrals. Mathematical Centre, Amsterdam.zbMATHGoogle Scholar
  13. Gu, M. G., Lai, T. L., and Lan, K. K. G. (1991). Rank tests based on censored data and their sequential analogues. Amer. J. Math. & Management Sci., 11(1–2):147–176.MathSciNetzbMATHGoogle Scholar
  14. Hajek, J. (1962). Asymptotically most powerful rank-order tests. Ann. Math. Statist., 33(3):1124–1147.MathSciNetCrossRefGoogle Scholar
  15. Hajek, J. (1968). Asymptotic normality of simple linear rank statistics under alternatives. Ann. Math. Statist., 39:325–346.MathSciNetCrossRefGoogle Scholar
  16. Hajek, J. (1970). A characterization of limiting distributions of regular estimates. Z. fur Wahrsch. und Verw. Gebiete, 14:323–330.MathSciNetCrossRefGoogle Scholar
  17. Hajek, J. (1972). Local asymptotic minimax and admissibility in estimation. In L. LeCam, J. N. and Scott, E., editors, Proc. Sixth Berkeley Symp. Math. Statist. Prob., volume 1, pages 175–194. University of California Press, Berkeley.Google Scholar
  18. Huber, P. J. (1964). Robust estimation of a location parameter. Annals of Mathematical Statistics, 35(1):73–101.MathSciNetCrossRefGoogle Scholar
  19. Huber, P. J. (1972). The 1972 Wald lecture robust statistics: A review. Annals of Mathematical Statistics, 43(4):1041–1067.MathSciNetCrossRefGoogle Scholar
  20. Huber, P. J. (1973). Robust regression: Asymptotics, conjectures and Monte Carlo. Annals of Statistics, 1(5):799–821.MathSciNetCrossRefGoogle Scholar
  21. Huber, P. J. (1981). Robust statistics. Wiley, New York.CrossRefGoogle Scholar
  22. Kalbfleisch, J. D. and Prentice, R. L. (1973). Marginal likelihoods based on cox’s regression and life model. Biometrika, 60(2):267–278.MathSciNetCrossRefGoogle Scholar
  23. Lai, T. L. and Ying, Z. (1991). Rank regression methods for left-truncated and right-censored data. Ann. Statist., 19(2):531–556.MathSciNetCrossRefGoogle Scholar
  24. Lai, T. L. and Ying, Z. (1992). Asymptotically efficient estimation in censored and truncated regression models. Statistica Sinica, 2(1):17–46.MathSciNetzbMATHGoogle Scholar
  25. Mantel, N. (1966). Evaluation of survival data and two new rank order statistics arising in its consideration. Cancer Chemotherapy Reports, 50(3):163–170.Google Scholar
  26. Prentice, R. (1978). Linear rank tests with right censored data. Biometrika, 65:167–179.MathSciNetCrossRefGoogle Scholar
  27. Rodriguez, G. (2005). Nonparametric Survival Models. Princeton University Press.Google Scholar
  28. van der Vaart, A. (2007). Asymptotic Statistics. Cambridge University Press.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Mayer Alvo
    • 1
  • Philip L. H. Yu
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada
  2. 2.Department of Statistics and Actuarial ScienceUniversity of Hong KongHong KongChina

Personalised recommendations