Optimal Boundary Kernels and Weightings for Local Polynomial Regression

TL;DR

This paper derives optimal boundary kernels for local polynomial regression that minimize mean square error, showing their equivalence to weighted polynomial fitting and providing a fast computation algorithm.

Contribution

It introduces optimal boundary kernels and weightings for local polynomial regression, establishing their equivalence and proposing an efficient computation method.

Findings

Optimal boundary kernels minimize mean square error.

Kernel estimators with certain moment conditions are equivalent to local polynomial regression.

A fast algorithm for boundary region kernel estimation is developed.

Abstract

Kernel smoothers are considered near the boundary of the interval. Kernels which minimize the expected mean square error are derived. These kernels are equivalent to using a linear weighting function in the local polynomial regression. It is shown that any kernel estimator that satisfies the moment conditions up to order $m$ is equivalent to a local polynomial regression of order $m$ with some non-negative weight function if and only if the kernel has at most $m$ sign changes. A fast algorithm is proposed for computing the kernel estimate in the boundary region for an arbitrary placement of data points.