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.

翻译：我们考虑两个样本积分泛函的估计问题，这类泛函自然出现在例如目标量为未知概率密度间散度的情形中。我们的首个主要结论是：在广泛的一般性条件下，加权最近邻估计量是有效的，即达到局部渐近极小化下界。此外，我们还证明了相应的中心极限定理，这有助于构建具有渐近最小宽度的泛函渐近有效置信区间。我们结果的一个有趣推论是：对于某些泛函，我们的估计量的最坏情况性能可能优于自然"预言机"估计量——后者可获取观测点处未知密度的真实值。