The Annals of Statistics

On Curve Estimation by Minimizing Mean Absolute Deviation and Its Implications

Jianqing Fan and Peter Hall

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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.

Article information

Ann. Statist., Volume 22, Number 2 (1994), 867-885.

First available in Project Euclid: 11 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties

Curve estimation kernel mean absolute deviation median minimax risk


Fan, Jianqing; Hall, Peter. On Curve Estimation by Minimizing Mean Absolute Deviation and Its Implications. Ann. Statist. 22 (1994), no. 2, 867--885. doi:10.1214/aos/1176325499.

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