Open Access
June, 1994 On Curve Estimation by Minimizing Mean Absolute Deviation and Its Implications
Jianqing Fan, Peter Hall
Ann. Statist. 22(2): 867-885 (June, 1994). DOI: 10.1214/aos/1176325499


The local median regression method has long been known as a robustified alternative to methods such as local mean regression. Yet, its optimal statistical properties are largely unknown. In this paper, we show via decision-theoretic arguments that a local weighted median estimator is the best least absolute deviation estimator in an asymptotic minimax sense, under $L_1$-loss. We also study asymptotic efficiency of the local median estimator in the class of all possible estimators. From a practical viewpoint our results show that local weighted medians are preferable to histogram estimators, since they enjoy optimality properties which the latter do not, under virtually identical smoothness assumptions on the underlying curve. Among smoothing methods that are adapted to functions with only one derivative, little is to be gained by using an estimator other than one based on the local median.


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Jianqing Fan. Peter Hall. "On Curve Estimation by Minimizing Mean Absolute Deviation and Its Implications." Ann. Statist. 22 (2) 867 - 885, June, 1994.


Published: June, 1994
First available in Project Euclid: 11 April 2007

zbMATH: 0806.62030
MathSciNet: MR1292544
Digital Object Identifier: 10.1214/aos/1176325499

Primary: 62G05
Secondary: 62G20

Keywords: Curve estimation , ‎kernel‎ , mean absolute deviation , median , minimax risk

Rights: Copyright © 1994 Institute of Mathematical Statistics

Vol.22 • No. 2 • June, 1994
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