Estimation of density functionals via cross‐validation

In density estimation, the mean integrated squared error (MISE) is commonly used as a measure of performance. In that setting, the cross‐validation criterion provides an unbiased estimator of the MISE minus the integral of the squared density. Since the minimum MISE is known to converge to zero, this suggests that the minimum value of the cross‐validation criterion could be regarded as an estimator of minus the integrated squared density. This novel proposal presents the outstanding feature that, unlike all other existing estimators, it does not need the choice of any tuning parameter. Indeed, it is proved here that this approach results in a consistent and efficient estimator, with remarkable performance in practice. Moreover, apart from this base case, it is shown how several other problems on density functional estimation can be similarly handled using this new principle, thus demonstrating full potential for further applications.