Reducing Variance in Smoothing

This paper proposes a variance reduction principle in nonparametric smoothing. At each point of estimation, Þrst obtain estimates at nearby points based on some pre- liminary procedure and then linearly combine the estimates to form the Þnal estimate. The linear combination is speciÞed to ensure that the asymptotic bias remains the same as that of the preliminary estimator. The nearby points are allocated in two asymmetric ways to facilitate as much asymptotic variance reduction as possible. An averaged version balances the second order biases and further improves variance. This technique is very general and has a wide range of applications. We study in details its application to the local linear regression estimator and examine the theoretical and numerical properties. While the new estimators retain many nice properties of the original local linear estimator, they achieve appealing asymptotic relative e�- ciencies. A simulation study indicates that the Þnite sample relative eciencies are usually compatible for moderate sample sizes. The proposed methods are very easy to implement, do not involve unknown factors, and bandwidth selection can be done by simple constant factor adjustments of those for local linear estimation. Coverage accuracy of conÞdence intervals based on the proposed estimators are investigated.