AI 中文总结
本文针对经典正态均值问题的交叉验证，证明了对偶随机化在光滑估计量下的最优性，给出了一类对偶方案并验证联合正态方案的极小极大最优性，还说明其对非光滑估计量的方差改进效果。
AI 中文摘要
在经典正态均值问题中，可通过用正态随机化扰动数据构建独立的训练-测试折，对K个此类折取平均得到交叉验证估计量，其偏差依赖于随机化变量的边际分布，方差则依赖于它们的联合分布。这引出两个问题：哪种联合分布是最优的，以及如何构造对应的随机化方案？本文证明：（i）对于光滑估计量，当偏差消失时，具有成对相关系数ρ=-1/(K-1)的对偶随机化是使随机化导致的可约方差保持有界的必要且充分条件；（ii）一种通用构造给出了一类对偶方案，其中联合正态方案是极小极大最优的；（iii）对于具有有限个跳跃间断点的非光滑估计量，对偶随机化可提升可约方差的渐近速率，而当间断点已知时，简单的控制变量可恢复有界方差。
英文摘要
In the classical normal means problem, independent train--test folds can be constructed by perturbing the data with normal randomization. Averaging over $K$ such folds yields a cross-validation estimator whose bias depends on the marginal distribution of the randomization variables, while its variance depends on their joint distribution. This raises the questions: which joint law is optimal, and how to construct the corresponding randomization scheme? We show that: (i) for smooth estimators, antithetic randomization with pairwise correlation $ρ=-1/(K-1)$ is necessary and sufficient for the reducible variance due to randomization to remain bounded as the bias vanishes; (ii) a general construction yields a class of antithetic schemes, within which the jointly normal scheme is minimax optimal; and (iii) for non-smooth estimators with finitely many jump discontinuities, antithetic randomization improves the asymptotic rate of the reducible variance, while a simple control variate restores bounded variance when the discontinuities are known.
Comments27 pages, 2 figures
