## The Annals of Statistics

### On Curve Estimation by Minimizing Mean Absolute Deviation and Its Implications

#### Abstract

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

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

Dates
First available in Project Euclid: 11 April 2007

https://projecteuclid.org/euclid.aos/1176325499

Digital Object Identifier
doi:10.1214/aos/1176325499

Mathematical Reviews number (MathSciNet)
MR1292544

Zentralblatt MATH identifier
0806.62030

JSTOR