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We consider the estimation of two-sample integral functionals, of the typethat occur naturally, for example, when the object of interest is a divergencebetween unknown probability densities. Our first main result is that, in widegenerality, a weighted nearest neighbour estimator is efficient, in the senseof achieving the local asymptotic minimax lower bound. Moreover, we also provea corresponding central limit theorem, which facilitates the construction ofasymptotically valid confidence intervals for the functional, havingasymptotically minimal width. One interesting consequence of our results is thediscovery that, for certain functionals, the worst-case performance of ourestimator may improve on that of the natural `oracle' estimator, which is givenaccess to the values of the unknown densities at the observations.

Efficient functional estimation and the super-oracle phenomenon