ON CURVE ESTIMATION BY MINIMIZING MEAN ABSOLUTE DEVIATION AND ITS IMPLICATIONS

• 成果类型: Article
• 署名作者: FAN, JQ; HALL, P
• 署名单位: Australian National University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176325499
• 发表日期: 1994
• 页码: 867-885
• 关键词: regression

摘要

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.