期刊
JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
卷 -, 期 -, 页码 -出版社
TAYLOR & FRANCIS INC
DOI: 10.1080/01621459.2022.2115375
关键词
Bagging; Bootstrap and jackknife; k-nearest neighbors; Nonparametric estimation and inference; Two-scale distributional nearest neighbors; Weighted nearest neighbors
资金
- NIH [1R01GM131407]
- NSF [DMS-1953356, EF-2125142, DMS-1953293]
This work introduces a novel two-scale DNN method by linearly combining two DNN estimators with different subsampling scales to reduce bias and achieve the optimal nonparametric convergence rate under the fourth-order smoothness condition.
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.
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