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Relative Errors of Difference-Based Variance Estimators in Nonparametric Regression

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Difference-based estimators for the error variance are popular since they do not require the estimation of the mean function. Unlike most existing difference-based estimators, new estimators proposed by Muller et al. (2003) and Tong and Wang (2005) achieved the asymptotic optimal rate as residual-based estimators. In this article, we study the relative errors of these difference-based estimators which lead to better understanding of the differences between them and residual-based estimators. To compute the relative error of the covariate-matched U-statistic estimator proposed by Muller et al. (2003), we develop a modified version by using simpler weights. We further investigate its asymptotic property for both equidistant and random designs and show that our modified estimator is asymptotically efficient.

Keywords: Asymptotically efficient; Bandwidth; Kernel estimator; Mean squared error; Nonparametric regression; Variance estimation

Document Type: Research Article

Affiliations: 1: Department of Applied Mathematics, University of Colorado, Boulder, Colorado, USA 2: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts, USA 3: Department of Statistics and Applied Probability, University of California, Santa Barbara, California, USA

Publication date: January 1, 2008

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