The Annals of Statistics

Local Linear Regression Smoothers and Their Minimax Efficiencies

Jianqing Fan

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In this paper we introduce a smooth version of local linear regression estimators and address their advantages. The MSE and MISE of the estimators are computed explicitly. It turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency-they are not only efficient in rates but also nearly efficient in constant factors. In the nonparametric regression context, the asymptotic minimax lower bound is developed via the heuristic of the "hardest onedimensional subproblem" of Donoho and Liu. Connections of the minimax risk with the modulus of continuity are made. The lower bound is also applicable for estimating conditional mean (regression) and conditional quantiles for both fixed and random design regression problems.

Article information

Ann. Statist., Volume 21, Number 1 (1993), 196-216.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G20: Asymptotic properties
Secondary: 62G05: Estimation 62F35: Robustness and adaptive procedures

Local linear smoothers hardest one-dimensional subproblem minimax risk modulus of continuity nonparametric regression


Fan, Jianqing. Local Linear Regression Smoothers and Their Minimax Efficiencies. Ann. Statist. 21 (1993), no. 1, 196--216. doi:10.1214/aos/1176349022.

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