The Maximal Smoothing Principle in Density Estimation
密度估计中的最大平滑原则AI 解读 ↓
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密度估计中的最大平滑原则
- 研究问题
- 非参数密度估计如何选择较稳定的平滑参数，在减少交叉验证参数波动的同时，控制过度平滑造成的信息损失？
- 方法与思路
- 作者主张采用与估计数据尺度相容的最大平滑量，用标准差及更稳健的尺度估计构造直方图和核方法的参数。
- 材料陈述的贡献
- 摘要称该原则降低平滑参数的抽样波动，并扩展至不同支持集及多变量情形，同时明确其保守而非渐近最优的性质。
- 阅读关注点
- 应核对尺度对最优平滑量给出上界的条件、不同尺度估计的作用，以及边界处理和辅助数据展示如何保留样本特征。
- 阅读边界
- OpenAlex索引摘要未注明稿件版本，变异阶的指数排版需核对原PDF。作者明确承认信息损失，不能据此认定其普遍优于交叉验证。
原文摘要
We propose a widely applicable method for choosing the smoothing parameters for nonparametric density estimators. It has come to be realized in recent years (e.g., see Hall and Marron 1987; Scott and Terrell 1987) that cross-validation methods for finding reasonable smoothing parameters from raw data are of very limited practical value. Their sampling variability is simply too large. The alternative discussed here, the maximal smoothing principle, suggests that we consider using the most smoothing that is consistent with the estimated scale of our data. This greatly generalizes and exploits a phenomenon noted in Terrell and Scott (1985), that measures of scale tend to place upper bounds on the smoothing parameters that minimize asymptotic mean integrated squared error of density estimates such as histograms and frequency polygons. The method avoids the extreme sampling variability of cross-validation by using ordinary scale estimators such as the standard deviation and interquartile range, which have order n −1 variability; cross-validated parameters have orders of variability such as n −1/5. The disadvantage is that maximal smoothing parameters are conservative, rather than asymptotically optimal. Because they tend to lose information, they should be used in conjunction with other data displays that retain more of the features of the original sample. On the other hand, such conservative methods are widely valued by statisticians because they discourage naive overinterpretation of one's data. Maximal smoothing parameters are here derived for histograms and kernel methods, using not only the standard deviation but several more resistant methods of scale estimation. The method is then applied to density estimation on the half-line, on finite intervals, and in several variables.