ON CURVE ESTIMATION BY MINIMIZING MEAN ABSOLUTE DEVIATION AND ITS IMPLICATIONS

The local median regression method has long been known as a robustifie d alternative to methods such as local mean regression. Yet, its optim al statistical properties are largely unknown. In this paper, we show via decision-theoretic arguments that a local weighted median estimato r is the best least absolute deviation estimator in an asymptotic mini max 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 prope rties which the latter do not, under virtually identical smoothness as sumptions on the underlying curve. Among smoothing methods that are ad apted to functions with only one derivative, little is to be gained by using an estimator other than one based on the local median.