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March, 1993 Local Linear Regression Smoothers and Their Minimax Efficiencies
Jianqing Fan
Ann. Statist. 21(1): 196-216 (March, 1993). DOI: 10.1214/aos/1176349022

Abstract

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.

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Jianqing Fan. "Local Linear Regression Smoothers and Their Minimax Efficiencies." Ann. Statist. 21 (1) 196 - 216, March, 1993. https://doi.org/10.1214/aos/1176349022

Information

Published: March, 1993
First available in Project Euclid: 12 April 2007

zbMATH: 0773.62029
MathSciNet: MR1212173
Digital Object Identifier: 10.1214/aos/1176349022

Subjects:
Primary: 62G20
Secondary: 62F35 , 62G05

Keywords: hardest one-dimensional subproblem , Local linear smoothers , minimax risk , modulus of continuity , Nonparametric regression

Rights: Copyright © 1993 Institute of Mathematical Statistics

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Vol.21 • No. 1 • March, 1993
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