Abstract
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 itself can be optimal in the related problem where the data consist of the values of the unknown densities at the observations.
Funding Statement
The first author was supported by Engineering and Physical Sciences Reseach Council (EPSRC) New Investigator Award EP/W016117/1.
The second author was supported in part by EPSRC Programme grant EP/N031938/1, EPSRC Fellowship EP/P031447/1 and European Research Council Advanced grant 101019498.
Acknowledgements
The authors are very grateful to the anonymous reviewers for their constructive comments, which helped to improve the paper.
Citation
Thomas B. Berrett. Richard J. Samworth. "Efficient functional estimation and the super-oracle phenomenon." Ann. Statist. 51 (2) 668 - 690, April 2023. https://doi.org/10.1214/23-AOS2265
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