Asymptotic comparison of (partial) cross-validation, GCV and randomized GCV in nonparametric regression
非参数回归中部分交叉验证、GCV 与随机化 GCV 的渐近比较AI 解读 ↓
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非参数回归中部分交叉验证、GCV 与随机化 GCV 的渐近比较
- 研究问题
- 非参数回归选择平滑参数时，随机化 GCV 和部分交叉验证分别牺牲多少统计表现，计算节省与超额误差之间如何比较？
- 方法与思路
- 作者比较各选择器相对最优平均平方误差的超额误差渐近分布，主要在核回归框架下研究，并允许非等距设计。
- 材料陈述的贡献
- 摘要称随机化造成的分布扩张因子小于一加随机迹估计次数的倒数，并指出特定比较下部分交叉验证的计算效率较低。
- 阅读关注点
- 应核对超额误差的标准化、分布扩张因子的含义、随机迹估计次数，以及部分验证比例超过一半时的等价比较条件。
- 阅读边界
- OpenAlex 摘要未注明稿件版本，比较结论主要针对核回归渐近框架；未提供有限样本运行时间，不能推广为普遍效率排序。
原文摘要
When using nonparametric estimates of the mean curve, surface or image underlying noisy observations, the selection of "smoothing parameters" is generally crucial. This paper gives a theoretical comparison of the performances of generalized cross-validation (GCV) and of its fast randomized version (RGCV), as selection criteria. This is mainly done by studying the asymptotic distribution of the excess error for each selector, that is, the difference between the (data-driven) resulting average squared error (ASE) and the best possible ASE. We show here that, by using randomization, this distribution is dilated, as compared to that for CV or GCV, only by a factor always lower than $1 + 1/n_R$, where $n_R$ is the number of primary randomized trace estimates one uses in RGCV. We include in the compared selectors, the partial cross-validation (PCV) approach where only a fraction of all the possible "leave-one-out" validation tests are evaluated; so that PCV is a common practice to reduce the computational cost in many contexts. In this paper, PCV will in fact appear as quite inefficient as compared to RGCV from this computational point of view. Moreover, we show that a precise comparison (and interpretation of the gain of using $n_R \geq 2$) is possible in terms of equivalent (in distribution) excess errors, if PCV uses a certain percentage of the test points greater than 50%. The obtained comparisons will be seen as quite reassuring on what is "sacrificed" in using randomized selectors. We give rigorous results mainly for the kernel regression setting as in the previous detailed study by Härdle, Hall and Marron of standard selectors, except that we do not restrict this one to an equidistant design.