A nonstandard application of cross-validation to estimate density functionals

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.