交叉验证：它估计什么以及估计效果如何？
Cross-validation: what does it estimate and how well does it do it?
AI总结：
该研究揭示交叉验证等方法实际估计的是同总体其他训练集模型的平均预测误差，指出常规交叉验证置信区间因折间相关覆盖率不足，提出嵌套交叉验证方案以更准确估计方差，提升区间覆盖率。
摘要：
交叉验证是一种被广泛用于估计预测误差的技术，但其行为复杂且尚未被完全理解。理想情况下，人们会认为交叉验证估计的是拟合于训练数据的当前模型的预测误差。我们证明，对于通过普通最小二乘法拟合的线性模型而言，事实并非如此；相反，它估计的是拟合于从同一总体中抽取的其他未见训练集的模型的平均预测误差。我们进一步表明，这种现象存在于大多数常用的预测误差估计方法中，包括数据分割、自助法（bootstrapping）以及Mallow's Cp。其次，由交叉验证推导的预测误差标准置信区间的覆盖率可能远低于预期水平。由于每个数据点同时被用于训练和测试，每一折的测量准确率之间存在相关性，因此常规的方差估计值过小。我们提出了一种嵌套交叉验证方案以更准确地估计该方差，并且我们通过实验证明，在许多传统交叉验证区间失效的示例中，这一改进所得到的区间具有近似正确的覆盖率。
英文摘要：
Cross-validation is a widely-used technique to estimate prediction error, but its behavior is complex and not fully understood. Ideally, one would like to think that cross-validation estimates the prediction error for the model at hand, fit to the training data. We prove that this is not the case for the linear model fit by ordinary least squares; rather it estimates the average prediction error of models fit on other unseen training sets drawn from the same population. We further show that this phenomenon occurs for most popular estimates of prediction error, including data splitting, bootstrapping, and Mallow's Cp. Next, the standard confidence intervals for prediction error derived from cross-validation may have coverage far below the desired level. Because each data point is used for both training and testing, there are correlations among the measured accuracies for each fold, and so the usual estimate of variance is too small. We introduce a nested cross-validation scheme to estimate this variance more accurately, and we show empirically that this modification leads to intervals with approximately correct coverage in many examples where traditional cross-validation intervals fail.
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