Bandwidth Selection for Kernel Density Estimation
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核密度估计的带宽选择
- 研究问题
- 最小二乘交叉验证带宽的样本波动较大，如何稳定自动带宽选择，并改善其收敛速度和有限样本表现？
- 方法与思路
- 作者通过特征函数分析交叉验证带宽的波动来源，提出稳定化选择器、插入式估计及调整后的插入式估计。
- 材料陈述的贡献
- 摘要称稳定化选择器强一致；密度足够光滑时相对收敛速度达平方根样本量级，并报告调整插入法与其渐近等价。
- 阅读关注点
- 应核对最优带宽的准则、不同速度所需的光滑条件、插入估计的效率含义，以及模拟中比较的密度与样本量。
- 阅读边界
- 仅据版本未注明的 OpenAlex 摘要；未给出具体光滑条件、实现细节和模拟数值，不能把所述速度或优势推广至任意密度。
原文摘要
The problem of automatic bandwidth selection for a kernel density estimator is considered. It is well recognized that the bandwidth estimate selected by the least squares cross-validation is subject to large sample variation. This difficulty limits the application of the cross-validation estimate. Based on characteristic functions, an important expression for the cross-validation bandwidth estimate is obtained. The expression clearly points out the source of variation. To stabilize the variation, a simple bandwidth selection procedure is proposed. It is shown that the stabilized bandwidth selector gives a strongly consistent estimate of the optimal bandwidth. Under commonly used smoothness conditions, the stabilized bandwidth estimate has a faster convergence rate than the convergence rate of the cross-validation estimate. For sufficiently smooth density functions, it is shown that the stabilized bandwidth estimate is asymptotically normal with a relative convergence rate $n^{-1/2}$ instead of the rate $n^{-1/10}$ of the cross-validation estimate. A plug-in estimate and an adjusted plug-in estimate are also proposed, and their asymptotic distributions are obtained. It is noted that the plug-in estimate is asymptotically efficient. The adjusted plug-in bandwidth estimate and the stabilized bandwidth estimate are shown to be asymptotically equivalent. The simulation results verify that the proposed procedures perform much better than the cross-validation for finite samples.