Journal of the Korean Statistical Society

The Korean Statistical Society (한국통계학회)
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A SIMPLE VARIANCE ESTIMATOR IN NONPARAMETRIC REGRESSION MODELS WITH MULTIVARIATE PREDICTORS

  • Lee Young-Kyung (Department of Statistics, Seoul National University) ;
  • Kim Tae-Yoon (Department of Statistics, Keimyung University) ;
  • Park Byeong-U. (Department of Statistics, Seoul National University)
  • Published : 2006.03.01
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Abstract

In this paper we propose a simple and computationally attractive difference-based variance estimator in nonparametric regression models with multivariate predictors. We show that the estimator achieves $n^{-1/2}$ rate of convergence for regression functions with only a first derivative when d, the dimension of the predictor, is less than or equal to 4. When d > 4, the rate turns out to be $n^{-4/(d+4)}$ under the first derivative condition for the regression functions. A numerical study suggests that the proposed estimator has a good finite sample performance.

Keywords

References

  1. BROWN, B. M. AND KILDEA, D. G. (1978). 'Reduced U-statistics and Hodges-Lehmann estimator', The Annals of Statistics, 6, 828-835 https://doi.org/10.1214/aos/1176344256
  2. CARROLL, R. J. (1988). 'The effects of variance function estimation on prediction and calibration: An example', In Statistical Decision Theory and Related Topics IV, Vol. 2 (J. O. Berger and S. S. Gupta, eds.), Springer-Verlag, New York
  3. CARROLL, R. J. AND RUPPERT, D. (1988). Transformation and Weighting in Regression, Chapman and Hall, New York
  4. DETTE, H., MUNK, A. AND WAGNER, T. (1998). 'Estimating the variance in nonparametric regression-what is a reasonable choice?', Journal of the Royal Statistical Society, Ser. B, 60, 751-764 https://doi.org/10.1111/1467-9868.00152
  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. AND MARRON, J. S. (1990). 'On variance estimation in nonparametric regression' , Biometrika. 77. 415-419 https://doi.org/10.1093/biomet/77.2.415
  8. 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
  9. HALL, P., KAY, J. W. AND TITTERINGTON, D. M. (1991). 'On estimation of noise variance in two-dimensional signal processing', Advances in Applied Probability, 23, 476-495 https://doi.org/10.2307/1427618
  10. KULASEKERA, K. B. AND GALLAGHER, C. (2002). 'Variance estimation in nonparametric multiple regression', Communications in Statistics-Theory and Methods, 31, 1373-1383 https://doi.org/10.1081/STA-120006074
  11. MULLER, H. G. AND STADTMULLER U. (1987). 'Estimation of heteroscedascity in regression analysis', The Annals of Statistics, 15, 610-625 https://doi.org/10.1214/aos/1176350364
  12. NEUMANN, M. H. (1994). 'Fully data-driven nonparametric variance estimators' Statistics, 25, 189-212 https://doi.org/10.1080/02331889408802445
  13. PARK, B. U., KIM, T. Y, LEE, Y K. AND PARK, C. (2006). 'A simple estimator of error correlation in nonparametric regression models', Scandinavian Journal of Statistics-Theory and Applications, in print
  14. RICE, J. (1984). 'Bandwidth choice for nonparametric regression', The Annals of Statistics, 12, 1215-1230 https://doi.org/10.1214/aos/1176346788
  15. SPOKOINY, V. (2002). 'Variance estimation for high-dimensional regression models', Journal of Multivariate Analysis, 82, 111-133 https://doi.org/10.1006/jmva.2001.2023