Local Bandwidth Selection for Kernel Estimates
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核估计的局部带宽选择
- 研究问题
- 如何按位置自适应选择非参数回归核估计的带宽，使局部均方误差得到改善，并刻画所选带宽的渐近性质？
- 方法与思路
- 作者先构造精确均方误差的一致估计，再最小化该估计选择局部带宽，并以模拟比较局部与全局选择程序。
- 材料陈述的贡献
- 摘要称，文章给出带宽渐近最优和渐近正态的充分条件，并报告局部程序在所用模拟中降低平均均方误差。
- 阅读关注点
- 应核对误差估计如何构造、搜索区间与充分条件，以及模拟基准；还应关注与渐近误差驱动方法的小样本比较。
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- 仅据 OpenAlex 索引摘要，稿件版本未注明，不能推断刊本变化；模拟改善依赖所用设计，摘要明确称与另一局部方法的小样本比较尚待研究。
原文摘要
A kernel estimate of a curve that uses an adaptive procedure for local selection of the bandwidth is considered here. A two-step procedure is proposed for estimating the local bandwidth that minimizes the mean squared error (MSE) of a kernel estimator for nonparametric regression. First, a consistent estimate of the exact MSE is constructed. Then the bandwidth that minimizes the estimate of the MSE is calculated. Sufficient conditions under which this bandwidth is asymptotically optimal and normally distributed are given. The local bandwidth selection procedure was implemented on some simulated data and compared to a global bandwidth selection procedure proposed by Rice (1984b). A 68%–91% reduction in the average MSE of a kernel estimator was realized with the local bandwidth selection procedure. Such a scheme was also studied by Müller (1985) and termed a direct pilot estimator approach. Müller derived sufficient conditions similar to those presented here, under which the direct pilot estimator approach provides a consistent estimator of the local bandwidth. His result is slightly more general in the specification of the interval that is searched for the optimizing bandwidth. The asymptotic normality of the optimizing bandwidth and the simulation results presented here have not to our knowledge appeared in the statistical literature. Müller and Stadtmüller (1987) proposed a local bandwidth selection procedure for kernel estimates that is based on the asymptotic expression for the MSE of the kernel estimator. Our procedure is based upon the finite sample expression for the MSE. It remains to be shown how the two procedures compare when applied to small data sets.