我们调查一般适用和直观吸引力的预测区间的基础上$k$倍交叉验证。我们专注于建议的区间的条件覆盖概率，给定训练样本中的观察结果（因此，训练条件有效性），并表明它在适当的意义上接近标称水平，前提是用于计算点预测的基础算法在省略特征-响应对时足够稳定。我们的研究结果是基于有限样本分析的经验分布函数的$k$倍交叉验证残差，并保持在非参数设置只有最小的假设上的误差分布。为了说明我们的结果，我们还将其应用于高维线性预测，在那里我们获得一致的渐近训练条件有效性，因为样本量和维度都以相同的速率趋于无穷大，并且一致的参数估计通常失败。这些结果表明，尽管存在着严重的问题，restling程序推断的未知参数（参见。Bickel和Freedman，1983; El Karoui和Purdom，2018; Mammen，1996），交叉验证方法可以成功地应用于获得可靠的预测推理，即使在高维和有条件的训练数据。
英文摘要：
We investigate generically applicable and intuitively appealing prediction intervals based on $k$-fold cross validation. We focus on the conditional coverage probability of the proposed intervals, given the observations in the training sample (hence, training conditional validity), and show that it is close to the nominal level, in an appropriate sense, provided that the underlying algorithm used for computing point predictions is sufficiently stable when feature-response pairs are omitted. Our results are based on a finite sample analysis of the empirical distribution function of $k$-fold cross validation residuals and hold in non-parametric settings with only minimal assumptions on the error distribution. To illustrate our results, we also apply them to high-dimensional linear predictors, where we obtain uniform asymptotic training conditional validity as both sample size and dimension tend to infinity at the same rate and consistent parameter estimation typically fails. These results show that despite the serious problems of resampling procedures for inference on the unknown parameters (cf. Bickel and Freedman, 1983; El Karoui and Purdom, 2018; Mammen, 1996), cross validation methods can be successfully applied to obtain reliable predictive inference even in high dimensions and conditionally on the training data.
Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong SAR, China（香港大学统计与精算科学系）；Musketeers Foundation Institute of Data Science, The University of Hong Kong, Hong Kong SAR, China（香港大学数据科学穆斯克特基金会研究所）；Department of Statistics and Data Science, Carnegie Mellon University, Pittsburgh, PA 15213, USA（卡内基梅隆大学统计与数据科学系）；Computational Biology Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA（卡内基梅隆大学计算生物学系）；Machine Learning Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA（卡内基梅隆大学机器学习系）
