The Annals of Statistics

Variable Bandwidth and Local Linear Regression Smoothers

Jianqing Fan and Irene Gijbels

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In this paper we introduce an appealing nonparametric method for estimating the mean regression function. The proposed method combines the ideas of local linear smoothers and variable bandwidth. Hence, it also inherits the advantages of both approaches. We give expressions for the conditional MSE and MISE of the estimator. Minimization of the MISE leads to an explicit formula for an optimal choice of the variable bandwidth. Moreover, the merits of considering a variable bandwidth are discussed. In addition, we show that the estimator does not have boundary effects, and hence does not require modifications at the boundary. The performance of a corresponding plug-in estimator is investigated. Simulations illustrate the proposed estimation method.

Article information

Ann. Statist., Volume 20, Number 4 (1992), 2008-2036.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G07: Density estimation
Secondary: 62G20: Asymptotic properties 62J99: None of the above, but in this section

Boundary effects local linear smoother mean squared error nonparametric regression optimalities variable bandwidth


Fan, Jianqing; Gijbels, Irene. Variable Bandwidth and Local Linear Regression Smoothers. Ann. Statist. 20 (1992), no. 4, 2008--2036. doi:10.1214/aos/1176348900.

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