Optimal cross-validation in density estimation with the $L^{2}$-loss
L²损失下密度估计的最优交叉验证AI 解读 ↓
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L²损失下密度估计的最优交叉验证
- 研究问题
- 在密度估计中，交叉验证如何分别用于风险估计、估计导向的模型选择与真实模型识别，测试集大小应如何确定？
- 方法与思路
- 作者为投影估计量的留p法风险估计推导闭式表达式，并分别分析风险估计和两类模型选择目标，辅以模拟实验。
- 材料陈述的贡献
- 摘要称，留一法在所研究的风险估计程序中最优；模型选择所需的测试集比例则随估计或识别目标及收敛速度变化。
- 阅读关注点
- 应核对最优性对应的风险准则、模型集合条件，以及不同收敛速度下测试集比例的要求，区分有限样本与渐近结论。
- 阅读边界
- 仅据OpenAlex索引摘要，稿件版本未注明；具体模型条件与模拟结果未提供，不能把所述最优性推广到任意估计量或推断刊本变化。
原文摘要
We analyze the performance of cross-validation (CV) in the density estimation framework with two purposes: (i) risk estimation and (ii) model selection. The main focus is given to the so-called leave-$p$-out CV procedure (Lpo), where $p$ denotes the cardinality of the test set. Closed-form expressions are settled for the Lpo estimator of the risk of projection estimators. These expressions provide a great improvement upon $V$-fold cross-validation in terms of variability and computational complexity. From a theoretical point of view, closed-form expressions also enable to study the Lpo performance in terms of risk estimation. The optimality of leave-one-out (Loo), that is Lpo with $p=1$, is proved among CV procedures used for risk estimation. Two model selection frameworks are also considered: estimation, as opposed to identification. For estimation with finite sample size $n$, optimality is achieved for $p$ large enough [with $p/n=o(1)$] to balance the overfitting resulting from the structure of the model collection. For identification, model selection consistency is settled for Lpo as long as $p/n$ is conveniently related to the rate of convergence of the best estimator in the collection: (i) $p/n\to1$ as $n\to+\infty$ with a parametric rate, and (ii) $p/n=o(1)$ with some nonparametric estimators. These theoretical results are validated by simulation experiments.