Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Asymptotics for a censored generalized linear model with unknown link function
Download PDF
Download PDF
  • Published: 23 August 2006

Asymptotics for a censored generalized linear model with unknown link function

  • Yanhua Wang1,
  • Shuyuan He1,
  • Lixing Zhu2,3 &
  • …
  • Kam C. Yuen4 

Probability Theory and Related Fields volume 138, pages 235–267 (2007)Cite this article

  • 203 Accesses

  • 9 Citations

  • Metrics details

Abstract

For censored response variable against projected co-variable, a generalized linear model with an unknown link function can cover almost all existing models under censorship. Its special cases include the accelerated failure time model with censored data. Such a model in the uncensored case is called the single-index model in econometrics. In this paper, we systematically study the asymptotic properties. We derive the central limit theorem and the law of the iterated logarithm for an estimator of the direction parameter. We also obtain the optimal convergence rate of an estimator of the unknown link function in the model.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Billingsley P. (1968) Convergence of Probability Measures. Wiley, New York

    MATH  Google Scholar 

  2. Carroll R.J., Fan J.Q., Gijbels I., Wand M.P. (1997) Generalized partially linear single-index model. J. Am. Stat. Assoc. 92, 477–489

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen K., Lo S.H. (1997) On the rate of uniform convergence of the product-limit estimator : Strong and weak laws. Ann. Stat. 25, 1050–1087

    Article  MATH  MathSciNet  Google Scholar 

  4. Daniela C., Jeffrey H., Lépold S. (2002) Automatic smoothing and estimation in single index poisson regression. J. Nonpanametr. Stat. 14, 307–323

    Article  MATH  Google Scholar 

  5. Delecroix M., Härdle W., Hristache M. (2003) Efficient estimation in conditional single index regression. J. Multi. Anal. 86, 213–226

    Article  MATH  Google Scholar 

  6. Fan J., Gijbels I. (1994) Censored regression: local linear approximations and their applications. J. Am. Stat. Assoc. 89, 560–570

    Article  MATH  MathSciNet  Google Scholar 

  7. Friedman J.H., Stuetzle W. (1981) Projection pursuit regression. J. Am. Stati. Assoc. 76, 817–823

    Article  MathSciNet  Google Scholar 

  8. Gill R.D. (1983) Large sample behavior of the product limit estimator on the whole line. Ann. Stat. 11, 49–58

    MATH  MathSciNet  Google Scholar 

  9. Gu M., Lai L. (1990) Functional laws of the iterated logarithm for the product limit estimator of a distribution function under random censorship or truncation. Ann. Probab. 18, 160–189

    MATH  MathSciNet  Google Scholar 

  10. Hall P. (1989) On projection pursuit regression. Ann. Statist. 17, 573–588

    MATH  MathSciNet  Google Scholar 

  11. Härdle W., Hall P., Ichimura H. (1993) Optimal smoothing in single-index models. Ann. Stat. 21, 157–178

    MATH  Google Scholar 

  12. Härdle W., Stoker T. (1989) Investigating smooth multiple regression by the method of averrage derivatives. J. Am. Stat. Assoc. 84, 986–995

    Article  MATH  Google Scholar 

  13. He S.Y., Yang G. (1998) Estimation of the truncation probability in the random truncation model. Ann. Stat. 26, 1011–1027

    Article  MATH  MathSciNet  Google Scholar 

  14. He S.Y., Wong H. The central limit theorem of linear regression model under right censorship. Sci. China Ser. A.2003, (to appear) (2003)

  15. He S.Y., Wang Y.H. (2004) The law of the iterated logarithm of the Kaplan-Meier integral and its application. Chin. Ann. Math. 25B: 199–206

    Article  MATH  MathSciNet  Google Scholar 

  16. Horowitz J., Härdle W. (1996) Direct Semiparametric estimation of single-index models with discrete covariates. J. Am. Stat. Assoc. 91, 1632–1640

    Article  MATH  Google Scholar 

  17. Huber P. (1985) Projection pursuit (with discussion). Ann. Statist. 13, 435–475

    MATH  MathSciNet  Google Scholar 

  18. Huh J., Park B.U. (2002) Likelihood-based local polynomial fitting for single-index models. J. Multi. Anal. 80, 302–321

    Article  MATH  MathSciNet  Google Scholar 

  19. Ichimura H. Estimation of single-index models. PhD dissertation, Department of. Economics, MIT (1987)

  20. Ichimura H. (1993) Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J. Econom 58, 71–120

    Article  MATH  MathSciNet  Google Scholar 

  21. Koul H., Susarla V., Van Ryzin J. (1981) Regression analysis with randomly right censored data. Ann. Stat. 9, 1276–1288

    MATH  MathSciNet  Google Scholar 

  22. Leurgans S. (1987) Linear models, random censoring, and synthetic data. Biometrika 74, 301–309

    Article  MATH  MathSciNet  Google Scholar 

  23. Li K.C., Duan N. (1989) Regression analysis under link violation. Ann. Stat. 17, 1009–1052

    MATH  MathSciNet  Google Scholar 

  24. Li K.C., Wang J.L., Chen C. (1999) Dimension reduction for censored regression data. Ann. Stat. 27, 1–23

    MATH  MathSciNet  Google Scholar 

  25. Stute W., Wang J.L. (1993) The strong law under random censorship. Ann. Stat. 21, 1591–1607

    MATH  MathSciNet  Google Scholar 

  26. Stute W. (1995) The central limit theorem under random censorship. Ann. Stat. 23, 422–439

    MATH  MathSciNet  Google Scholar 

  27. Stute W. (1999). Nonlinear censored regression. Stat. Sinica91089–1102 (1999)

    Google Scholar 

  28. Wang Y.H., He S.Y., Zhu L.X., Yuen K.C.:Asymptotics for a single-index model under random censorship. Research Report Serial No.354, Department of Statistics and Actuarial Science, the University of Hong Kong (2005)

  29. Xia Y., Tong H., Li W.K. (1999) On extended partially linear single-index models. Biometrika 86, 831–842

    Article  MATH  MathSciNet  Google Scholar 

  30. Zhou M. (1991) Some properties of the Kaplan Meier estimator for indenpendent nonidentically distributed random variables. Ann. Stat. 19, 2266–2274

    MATH  Google Scholar 

  31. Zhu L.X., Fang K.T. (1992) On projection pursuit approximation for nonparametric regression.In: Proceedings. of Order Statistic. and Nonparametrics: Theory and Applications, pp. 455–469

Download references

Author information

Authors and Affiliations

  1. Peking University, Beijing, China

    Yanhua Wang & Shuyuan He

  2. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Road, Hong Kong, China

    Lixing Zhu

  3. East China Normal University, Shanghai, China

    Lixing Zhu

  4. The University of Hong Kong, Hong Kong, China

    Kam C. Yuen

Authors
  1. Yanhua Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shuyuan He
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lixing Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kam C. Yuen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lixing Zhu.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Wang, Y., He, S., Zhu, L. et al. Asymptotics for a censored generalized linear model with unknown link function. Probab. Theory Relat. Fields 138, 235–267 (2007). https://doi.org/10.1007/s00440-006-0022-5

Download citation

  • Received: 16 October 2004

  • Revised: 24 May 2006

  • Published: 23 August 2006

  • Issue Date: May 2007

  • DOI: https://doi.org/10.1007/s00440-006-0022-5

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Central limit theorem
  • Projection pursuit regression
  • Random censorship
  • Generalized linear model
  • Unknown link

Mathematics Subject Classification (2000)

  • Primary 62H99
  • Secondary 62H05
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature