Skip to main content
Log in

A Semi-Parametric Mode Regression with Censored Data

  • Published:
Mathematical Methods of Statistics Aims and scope Submit manuscript

Abstract

In this work we suppose that the random vector (X, Y) satisfies the regression model Y = m(X) + ϵ, where m(·) belongs to some parametric class {\({m_\beta}(\cdot):\beta \in \mathbb{K}\)} and the error ϵ is independent of the covariate X. The response Y is subject to random right censoring. Using a nonlinear mode regression, a new estimation procedure for the true unknown parameter vector β0is proposed that extends the classical least squares procedure for nonlinear regression. We also establish asymptotic properties for the proposed estimator under assumptions of the error density. We investigate the performance through a simulation study.

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

Similar content being viewed by others

References

  1. M. G. Akritas, “Nearest Neighbor Estimation of a Bivariate Distribution under Random Censoring”, Ann. Statist. 22, 1299–1327 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  2. M. G. Akritas, “On the Use of Nonparametric Regression Techniques for Fitting Parametric Regression Models”, Biometrics 52, 1342–1362 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Buckley and I. James, “Linear Regression with Censored Data”, Biometrika 66, 429–436 (1979).

    Article  MATH  Google Scholar 

  4. P. Deheuvels and J. H. J. Einmahl, “Functional Limit Laws for the Increments of Kaplan-Meier Product-Limit Processes and Applications”, Ann. Probab. 28, 1301–1335 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  5. R. M. Dudley, Uniform Central Limit Theorems (Cambridge Univ. Press, Cambridge, UK, 1999).

    Book  MATH  Google Scholar 

  6. W. F. Eddy, “Optimum Kernel Estimators of the Mode”, Ann. Statist. 8, 870–882 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  7. T. Gasser and H. G. Müller, “Estimating Regression Functions and Their Derivatives by the Kernel Method”, Scand. J. Statist. 11, 171–185 (1984).

    MathSciNet  MATH  Google Scholar 

  8. E. Giné and A. Guillou, “Rates of Strong Uniform Consistency for Multivariate Kernel Density Estimators”, Ann. Inst. Henri Poincare 38, 907–921 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  9. W. Härdle, H. Liang, and J. Gao, Partially Linear Models (Physica-Verlag, Heidelberg, 2000).

    Book  MATH  Google Scholar 

  10. R. Jennrich, “Asymptotic Properties of Nonlinear Least-Squares Estimator”, Ann. Math. Statist. 40, 633–643 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  11. E. L. Kaplan and P. Meier, “Nonparametric Estimation from Incomplete Observations”, J.Amer. Statist. Assoc. 53, 457–481 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Khardani, M. Lemdani, and E. Ould Saïd, “Some Asymptotic Properties for a Smooth Kernel Estimator of the Conditional Mode under Random Censorship”, J. Korean Statist. Soc. 39, 455–469 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Khardani, M. Lemdani, and E. Ould Said, “Uniform Rate of Strong Consistency for a Smooth Kernel Estimator of the Conditional Mode for Censored Time Series”, J. Statist. Plann. Inference 141, 3426–3436 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  14. S. Khardani, M. Lemdani, and E. Ould Saïd, “On the Strong Uniform Consistency of the Mode Estimator for Censored Time Series”, Metrika 135,1–13 (2012).

    MathSciNet  MATH  Google Scholar 

  15. S. Khardani and A. F. Yao, “Nonlinear Parametric Mode Regression”, Commun. Statist. Theory and Methods 46, 3006–3024 (2017).

    Article  MATH  Google Scholar 

  16. H. Koul, V. Susarla, and J. Van Ryzin, “Regression Analysis with Randomly Right-Censored Data”, Ann. Statist. 9, 1276–1288 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  17. M. J. Lee, “Mode Regression”, J. Econometric 42, 337–349 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  18. M. J. Lee, “Quadratic Mode Regression”, J. Econometric 57, 1–19 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Leurgans, “Linear Models, Random Censoring and Synthetic Data”, Biometrika 74, 301–309 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  20. H. Liang and W. Härdle, Asymptotic normality of parametric part in partially linear heteroscedastic regression models (DP 33, SFB 373, Humboldt Univ. Berlin).

  21. H. Liang, W. Härdle, and R. Carroll, “Estimation in a Semi-Parametric Partially Linear Errors in Variables Model”, Ann. Statist. 27, 1519–1535 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  22. R. G. Müller, “Least Squares Regression with Censored Data”, Biometrika 63, 449–464 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  23. E. Parzen, “On Estimation of a Probability Density Function and Mode”, Ann. Math. Statist. 33, 1065–1076 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  24. D. Ruppert, M. P. Wand, and R. J. Carroll, Semi-Parametric Regression (Cambridge Univ. Press, Cambridge, 2003).

    Book  MATH  Google Scholar 

  25. T. A. Severini and J. G. Staniswalis, “Quasi-Likelihood Estimation in Semi-Parametric Models”, J. Amer. Statist. Assoc. 89, 501–511 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  26. J. Shoung and C. Zhang, “Least Squares Estimators of the Mode of a Unimodal Regression Function”, Ann. Statist. 29, 648–665 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  27. W. Stute, “Consistent Estimation under Random Censorship when Covariates are Present”, J. Multivar. Anal. 45, 89–103 (1993).

    Article  MATH  Google Scholar 

  28. M. Talagrand, “New Concentration Inequalities in Product Spaces”, Invent. Math. 126, 505–563 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  29. I. Van Keilegom and M. G. Akritas, “The Least Squares Method in Heteroscedastic Censored Regression Models”, in: Asymptotics in Statistics and Probability Ed. by M. L. Puri (2000), pp. 379–391.

    Chapter  Google Scholar 

  30. A. W. Van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes with Applications to Statistics (Springer-Verlag, New York, 1996).

    Book  MATH  Google Scholar 

  31. W. Yao and L. Li, “A New Regression Model: Modal Linear Regression”, Scand. J. Statist. 41, 656–671 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Yatchew, Semi-Parametric Regression for the Applied Econometric (Cambridge Univ. Press, Cambridge, 2003).

    Book  MATH  Google Scholar 

  33. A. Yatchew, “An Elementary Estimator of the Partial Linear Model”, Economics Lett. 57, 135–143 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  34. M. Zhou, “Asymptotic Normality of the Synthetic Data Regression Estimator for Censored Survival Data”, Ann. Statist. 20, 1002–1021 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  35. K. Ziegler, “On the Asymptotic Normality of Kernel Regression Estimators of the Mode in the Random Design Model”, J. Statist. Plann. Inference 115, 123–144 (2003).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to S. Khardani.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khardani, S. A Semi-Parametric Mode Regression with Censored Data. Math. Meth. Stat. 28, 39–56 (2019). https://doi.org/10.3103/S1066530719010034

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066530719010034

Keywords

AMS 2010 Subject Classification

Navigation