Efficient functional estimation and the super-oracle phenomenon
This work addresses a fundamental statistical estimation problem for researchers in machine learning and statistics, offering an efficient method with theoretical guarantees.
The paper tackles the problem of estimating two-sample integral functionals, such as divergences between unknown probability densities, by proving that a weighted nearest neighbor estimator achieves the local asymptotic minimax lower bound and provides asymptotically valid confidence intervals with minimal width.
We consider the estimation of two-sample integral functionals, of the type that occur naturally, for example, when the object of interest is a divergence between unknown probability densities. Our first main result is that, in wide generality, a weighted nearest neighbour estimator is efficient, in the sense of achieving the local asymptotic minimax lower bound. Moreover, we also prove a corresponding central limit theorem, which facilitates the construction of asymptotically valid confidence intervals for the functional, having asymptotically minimal width. One interesting consequence of our results is the discovery that, for certain functionals, the worst-case performance of our estimator may improve on that of the natural `oracle' estimator, which is given access to the values of the unknown densities at the observations.