EFFICIENT FUNCTIONAL ESTIMATION AND THE SUPER-ORACLE PHENOMENON

• 成果类型: Article
• 署名作者: Berrett, Thomas B.; Samworth, Richard J.
• 署名单位: University of Warwick; University of Cambridge
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/23-AOS2265
• 发表日期: 2023
• 页码: 668-690
• 关键词: minimax density-estimation; integral functionals; ENTROPY ESTIMATION; distributions

摘要

We consider the estimation of two-sample integral functionals, of the type that occur naturally, for example, when the object of interest is a diver-gence 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 con-struction of asymptotically valid confidence intervals for the functional, hav-ing 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 it-self can be optimal in the related problem where the data consist of the values of the unknown densities at the observations.