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.Comment: 82 page