通过$k$ -最近邻进行分布函数的系综估计
Ensemble Estimation of Distributional Functionals via $k$-Nearest Neighbors
摘要：
分布泛函（一个或多个概率分布的积分泛函）的精确非参数估计问题由于其在信号处理、信息论、机器学习和统计学中的广泛适用性而最近受到关注。特别地，基于$k$ -最近邻（nn）的方法由于它们的自适应性质和它们相对低的计算复杂性而受到很多关注。我们得到了一类分布泛函的留一法$k$ -nn插入式密度估计的均方误差（MSE）收敛速度。然后，我们应用最优加权集成估计的理论，以获得加权集成估计，实现参数MSE率的假设下，是有竞争力的最先进的。这些估计，这是未知的所有其他$k$ -nn基于分布函数估计的渐近分布，也提出了这使我们能够进行假设检验。
英文摘要：
The problem of accurate nonparametric estimation of distributional functionals (integral functionals of one or more probability distributions) has received recent interest due to their wide applicability in signal processing, information theory, machine learning, and statistics. In particular, $k$-nearest neighbor (nn) based methods have received a lot of attention due to their adaptive nature and their relatively low computational complexity. We derive the mean squared error (MSE) convergence rates of leave-one-out $k$-nn plug-in density estimators of a large class of distributional functionals without boundary correction. We then apply the theory of optimally weighted ensemble estimation to obtain weighted ensemble estimators that achieve the parametric MSE rate under assumptions that are competitive with the state of the art. The asymptotic distributions of these estimators, which are unknown for all other $k$-nn based distributional functional estimators, are also presented which enables us to perform hypothesis testing.
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