Acta Mathematicae Applicatae Sinica, English Series

, Volume 19, Issue 3, pp 387–396 | Cite as

Coverage Accuracy of Confidence Intervals in Nonparametric Regression

Original papers

Abstract

Point-wise confidence intervals for a nonparametric regression function with random design points are considered. The confidence intervals are those based on the traditional normal approximation and the empirical likelihood. Their coverage accuracy is assessed by developing the Edgeworth expansions for the coverage probabilities. It is shown that the empirical likelihood confidence intervals are Bartlett correctable.

Keywords

Confidence interval empirical likelihood Nadaraya-Watson estimator normal approximation 

2000 MR Subject Classification

62G05 62E20 

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References

  1. 1.
    Chen, S.X. On the coverage accuracy of empirical likelihood confidence regions for linear regression model. Ann. Inst. Statist. Math., 45:621–637 (1993)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chen, S.X. Empirical likelihood confidence intervals for linear regression coefficients. J. Mult. Anal., 49:24–40 (1994)CrossRefGoogle Scholar
  3. 3.
    Chen, S.X. Empirical likelihood for nonparametric density estimation. Biometrika, 83:329–341 (1996)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen, S.X., Hall, P. Smoothed empirical likelihood confidence intervals for quantiles. Ann. Statist., 21:1166–1181 (1993)MathSciNetCrossRefGoogle Scholar
  5. 5.
    DiCiccio, T.J., Hall, P., Romano, J.P. Bartlett adjustment for empirical likelihood. Ann. Statist., 19:1053–1061 (1991)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fukunaga, K. Introduction to Statistical Pattern Recognition. New York:Academic Press, 1972Google Scholar
  7. 7.
    Hall, P. Edgeworth expansions for nonparametric density estimators, with applications. Statistics, 22:215–232 (1991)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hall, P. On bootstrap confidence intervals in nonparametric regression. Ann. Statist., 20:695–711 (1992)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hall, P. The bootstrap and Edgeworth Expansions. New York:Springer–Verlag, 1992Google Scholar
  10. 10.
    Hall, P., La Scala, B. Methodology and algorithms of empirical likelihood. Internat. Statist. Rev., 58:109–127 (1990)CrossRefGoogle Scholar
  11. 11.
    Härdle, W. Applied nonparametric regression. Cambridge:Cambridge University Press, 1990Google Scholar
  12. 12.
    James, G.S., Mayne, A.J. Cumulants of functions of random variables. Sankhya (Ser.A) 24:47–54 (1962)Google Scholar
  13. 13.
    Jing, B.Y., Wood, A.T.A. Exponential empirical likelihood is not Bartlett correctable. Ann. Statist., 24:365–369 (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Owen, A. Empirical likelihood ratio confidence intervals for a single functional. Biometrika, 75:237–249 (1988)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Owen, A. Empirical likelihood ratio confidence regions. Ann. Statist., 18:90–120 (1990)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Press, W.H., Flannery, B.F., Teukolsky, S.A., Vetterling, W.T. Numerical recipes in C:the art of scientific computing. Cambridge:Cambridge University Press, 1992)Google Scholar
  17. 17.
    Schuster, E.F. Joint asymptotic distribution of the estimated regression function at a finite number of distinct points. Ann. Math. Statist., 43:84–88 (1972)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Department of Statistics and Applied ProbabilityNational University of SingaporeSingaporeSingapore
  2. 2.Department of MathematicsGuangxi Normal UniversityGuilinChina

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