  The weighted nearest neighbors (WNN) estimator has been popularly used as a
flexible and easy-to-implement nonparametric tool for mean regression
estimation. The bagging technique is an elegant way to form WNN estimators with
weights automatically generated to the nearest neighbors; we name the resulting
estimator as the distributional nearest neighbors (DNN) for easy reference.
Yet, there is a lack of distributional results for such estimator, limiting its
application to statistical inference. Moreover, when the mean regression
function has higher-order smoothness, DNN does not achieve the optimal
nonparametric convergence rate, mainly because of the bias issue. In this work,
we provide an in-depth technical analysis of the DNN, based on which we suggest
a bias reduction approach for the DNN estimator by linearly combining two DNN
estimators with different subsampling scales, resulting in the novel two-scale
DNN (TDNN) estimator. The two-scale DNN estimator has an equivalent
representation of WNN with weights admitting explicit forms and some being
negative. We prove that, thanks to the use of negative weights, the two-scale
DNN estimator enjoys the optimal nonparametric rate of convergence in
estimating the regression function under the fourth-order smoothness condition.
We further go beyond estimation and establish that the DNN and two-scale DNN
are both asymptotically normal as the subsampling scales and sample size
diverge to infinity. For the practical implementation, we also provide variance
estimators and a distribution estimator using the jackknife and bootstrap
techniques for the two-scale DNN. These estimators can be exploited for
constructing valid confidence intervals for nonparametric inference of the
regression function. The theoretical results and appealing finite-sample
performance of the suggested two-scale DNN method are illustrated with several
numerical examples.
