On bias reduction in local linear smoothing
On bias reduction in local linear smoothing
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DOI：
10.1093/biomet/85.2.333
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发表时间：
1998-06
期刊：
影响因子：
2.7
通讯作者：
E. Choi;P. Hall
中科院分区：
文献类型：
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作者：
E. Choi;P. Hall
摘要 局部线性回归的标准方法涉及以对称方式将直线段拟合到曲线，因为该段直接拟合在一个小区域的上方，该区域的中点是横坐标 x，我们希望在该区域估计曲线。在本文中，我们表明，如果该段以倾斜方式拟合，其中心稍微偏向 x 的左侧或右侧，则偏差可以减少一个数量级，而不影响方差的数量级。中心应该移动的量仅取决于核函数，而不取决于未知的回归平均值或设计密度。两个相似但方向相反的偏移估计量的平均值的偏差减少了两个数量级，同样以方差略有增加为代价。该特定估计器可以被视为三个局部线性估计器（两个相反移位估计器和对称估计器）的凸组合的限制形式，其偏差小两个数量级，并且根据核函数，方差也更小。
SUMMARY The standard approach to local linear regression involves fitting a straight line segment to a curve in a symmetrical way, in that the segment is fitted directly above a small region whose midpoint is the abscissa, x, at which we wish to estimate the curve. In this paper we show that, if the segment is fitted in a skew manner, with its centre a little to the left or right of x, then bias can be reduced by an order of magnitude, without affecting the order of magnitude of variance. The amount by which the centre should be shifted depends only on the kernel function, and not at all on the unknown regression mean or on the design density. The average of two similarly but oppositely shifted estimators has two orders of magnitude less bias, again at the expense of a slight increase in variance. This particular estimator may be viewed as a limiting form of a convex combination of three local linear estimators, the two oppositely shifted estimators and the symmetric one, which has two orders of magnitude less bias and, depending on the kernel function, less variance as well.