ON CURVE ESTIMATION BY MINIMIZING MEAN ABSOLUTE DEVIATION AND ITS IMPLICATIONS
ON CURVE ESTIMATION BY MINIMIZING MEAN ABSOLUTE DEVIATION AND ITS IMPLICATIONS
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DOI：
10.1214/aos/1176325499
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发表时间：
1994-06-01
影响因子：
4.5
通讯作者：
HALL, P
中科院分区：
文献类型：
--
作者：
FAN, JQ;HALL, P
局部中值回归方法长期以来一直被认为是局部均值回归等方法的鲁棒替代方法。然而，其最佳统计特性在很大程度上是未知的。在本文中，我们通过决策理论的论证，局部加权中位数估计是最好的最小绝对偏差估计在渐近极小极大意义下，在L（1）-损失。我们还研究了局部中位估计在所有可能的估计类中的渐近有效性。从一个实际的角度来看，我们的研究结果表明，局部加权中位数是更可取的直方图估计，因为他们享有最优性能，后者没有，在基本曲线上几乎相同的平滑假设。在适应于只有一个导数的函数的平滑方法中，通过使用基于局部中值的估计器之外的估计器几乎没有获得。
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.