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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)
Publication Information
Journal of the Korean Statistical Society / v.35, no.1, 2006 , pp. 105-114 More about this Journal
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
Variance estimation; multivariate regression; rate of convergence;
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1 CARROLL, R. J. AND RUPPERT, D. (1988). Transformation and Weighting in Regression, Chapman and Hall, New York
2 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
3 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   DOI   ScienceOn
4 MULLER, H. G. AND STADTMULLER U. (1987). 'Estimation of heteroscedascity in regression analysis', The Annals of Statistics, 15, 610-625   DOI
5 NEUMANN, M. H. (1994). 'Fully data-driven nonparametric variance estimators' Statistics, 25, 189-212   DOI
6 GASSER, T., SROKA, L. AND JENNEN-STEINMETZ, C. (1986). 'Residual variance and residual pattern in nonlinear regression', Biometrika, 73, 625-633   DOI   ScienceOn
7 HALL, P. AND MARRON, J. S. (1990). 'On variance estimation in nonparametric regression' , Biometrika. 77. 415-419   DOI   ScienceOn
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   DOI   ScienceOn
9 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
10 RICE, J. (1984). 'Bandwidth choice for nonparametric regression', The Annals of Statistics, 12, 1215-1230   DOI
11 SPOKOINY, V. (2002). 'Variance estimation for high-dimensional regression models', Journal of Multivariate Analysis, 82, 111-133   DOI   ScienceOn
12 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   DOI   ScienceOn
13 KULASEKERA, K. B. AND GALLAGHER, C. (2002). 'Variance estimation in nonparametric multiple regression', Communications in Statistics-Theory and Methods, 31, 1373-1383   DOI   ScienceOn
14 BROWN, B. M. AND KILDEA, D. G. (1978). 'Reduced U-statistics and Hodges-Lehmann estimator', The Annals of Statistics, 6, 828-835   DOI
15 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