Journal of the Korean Statistical Society

  • 제35권1호
  • /
  • Pages.1-23
  • /
  • 2006
  • /
  • 1226-3192(pISSN)
  • /
  • 2005-2863(eISSN)
한국통계학회 (The Korean Statistical Society)
권호 표지

NONPARAMETRIC ESTIMATION OF THE VARIANCE FUNCTION WITH A CHANGE POINT

  • Kang Kee-Hoon (Department of Statistics, Hankuk University of Foreign Studies) ;
  • Huh Jib (Department of Statistics, Duksung Women's University)
  • 발행 : 2006.03.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper we consider an estimation of the discontinuous variance function in nonparametric heteroscedastic random design regression model. We first propose estimators of the change point in the variance function and then construct an estimator of the entire variance function. We examine the rates of convergence of these estimators and give results for their asymptotics. Numerical work reveals that using the proposed change point analysis in the variance function estimation is quite effective.

키워드

참고문헌

  1. BHATTACHARYA, P. K. AND BROCKWELL, P. J. (1976). 'The minimum of an additive process with applications to signal estimation and storage theory', Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 37, 51-75 https://doi.org/10.1007/BF00536298
  2. BILLINGSLEY, P. (1968). Convergence of Probability Measures. John Wiley & Sons, New York
  3. CARROLL, R. J. (1982). 'Adapting for heteroscedasticity in linear models', The Annals of Statistics, 10, 1224-1233 https://doi.org/10.1214/aos/1176345987
  4. CARROLL, R. J. AND RUPPERT, D. (1988). Transformation and Weighting in Regression. Chapman and Hall, London
  5. GASSER, T., SROKA, L. AND JENNEN-STEINMETZ, C. (1986). 'Residual variance and residual pattern in nonlinear regression', Biometrika, 73, 625-633 https://doi.org/10.1093/biomet/73.3.625
  6. HALL, P. AND CARROLL, R. J. (1989). 'Variance function estimation in regression: The effect of estimating the mean', Journal of the Royal Statistical Society, Ser. B, 51, 3-14
  7. HALL, P., KAY, J. W. AND TITTERINGTON, D. M. (1990). 'Asymptotically optimal difference-based estimation of variance in nonparametric regression', Biometrika, 77, 521-528 https://doi.org/10.1093/biomet/77.3.521
  8. HARDLE, W. (1990). Applied Nonparametric Regression. Cambridge University Press, Cambridge
  9. HARDLE, W. (1991). Smoothing Techniques, With Implementation in S. Springer-Verlag, New York
  10. LOADER, C. R. (1996). 'Change point estimation using nonparametric regression', The Annals of Statistics, 24, 1667-1678 https://doi.org/10.1214/aos/1032298290
  11. MACK, Y. P. AND SILVERMAN, B. W. (1982). 'Weak and strong uniform consistency of kernel regression estimates', Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 61, 405-415 https://doi.org/10.1007/BF00539840
  12. MULLER, H.-G. (1988). Nonparametric Regression Analysis of Longitudinal Data. SpringerVerlag, Berlin
  13. MULLER, H.-G. (1992). 'Change-points in nonparametric regression analysis', The Annals of Statistics, 20, 737-761 https://doi.org/10.1214/aos/1176348654
  14. MULLER, H.-G. AND STADTMULLER, U. (1987). 'Estimation of heteroscedasticity in regression analysis', The Annals of Statistics, 15, 610-625 https://doi.org/10.1214/aos/1176350364
  15. RAIMONDO, M. (1998). 'Minimax estimation of sharp change points', The Annals of Statistics, 26, 1379-1397 https://doi.org/10.1214/aos/1024691247
  16. RICE, J. (1984). 'Bandwidth choice for nonparametric regression', The Annals of Statistics, 12, 1215-1230 https://doi.org/10.1214/aos/1176346788
  17. RUPPERT, D., WAND, M. P., HOLST, U. AND HOSSJER, O. (1997). 'Local polynomial variance-function estimation', Technomtrics, 39, 262-273 https://doi.org/10.2307/1271131
  18. STUTE, W. (1982). 'A law of the logarithm for kernel density estimators', The Annals of Probability, 10, 414-422 https://doi.org/10.1214/aop/1176993866
  19. WHITT, W. (1970). 'Weak convergence of probability measures on the function space C[0, $\infty$)', Annals of Mathematical Statistics, 41, 939-944 https://doi.org/10.1214/aoms/1177696970