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April 2008 Effect of mean on variance function estimation in nonparametric regression
Lie Wang, Lawrence D. Brown, T. Tony Cai, Michael Levine
Ann. Statist. 36(2): 646-664 (April 2008). DOI: 10.1214/009053607000000901


Variance function estimation in nonparametric regression is considered and the minimax rate of convergence is derived. We are particularly interested in the effect of the unknown mean on the estimation of the variance function. Our results indicate that, contrary to the common practice, it is not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean when the mean function is not smooth. Instead it is more desirable to use estimators of the mean with minimal bias. On the other hand, when the mean function is very smooth, our numerical results show that the residual-based method performs better, but not substantial better than the first-order-difference-based estimator. In addition our asymptotic results also correct the optimal rate claimed in Hall and Carroll [J. Roy. Statist. Soc. Ser. B 51 (1989) 3–14].


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Lie Wang. Lawrence D. Brown. T. Tony Cai. Michael Levine. "Effect of mean on variance function estimation in nonparametric regression." Ann. Statist. 36 (2) 646 - 664, April 2008.


Published: April 2008
First available in Project Euclid: 13 March 2008

zbMATH: 1133.62033
MathSciNet: MR2396810
Digital Object Identifier: 10.1214/009053607000000901

Primary: 62G08 , 62G20

Keywords: minimax estimation , Nonparametric regression , variance estimation

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.36 • No. 2 • April 2008
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