Cross-validation is usually employed to evaluate the performance of a given statistical methodology. When such a methodology depends on a number of tuning parameters, cross-validation proves to be helpful to select the parameters that optimize the estimated performance. In this paper, however, a very different and nonstandard use of cross-validation is investigated. Instead of focusing on the cross-validated parameters, the main interest is switched to the estimated value of the error criterion at optimal performance. It is shown that this approach is able to provide consistent and efficient estimates of some density functionals, with the noteworthy feature that these estimates do not rely on the choice of any further tuning parameter, so that, in that sense, they can be considered to be purely empirical. Here, a base case of application of this new paradigm is developed in full detail, while many other possible extensions are hinted as well.

翻译：交叉验证通常用于评估给定统计方法的性能。当此类方法依赖于多个调优参数时，交叉验证可帮助选择优化估计性能的参数。然而，本文探讨了一种截然不同且非标准的交叉验证应用方式。研究重点并非关注交叉验证后的参数，而是转向误差准则在最优性能下的估计值。结果表明，该方法能够对某些密度泛函提供一致且有效的估计，其显著特征在于这些估计不依赖于任何额外调优参数的选择，因此在某种意义上可被视为纯经验性估计。本文详细阐述了这一新范式的基础应用案例，并同时提出了多种可能的扩展方向。