Local bandwidth selection via second derivative segmentation

Abstract

This paper studies the problem of local bandwidth selection for local linear regression. It is known that the optimal local bandwidth for estimating the unknown curve f at design point x depends on the curve’s second derivative f''(x) at x. Therefore one could select the local bandwidth h(x) at x via estimating f''(x). However, as typically estimating f''(x) is a much harder task than estimating f(x) itself, this approach for choosing h(x) tends to produce less accurate results. This paper proposes a method for choosing h(x) that bypasses the estimation of f''(x), yet at the same time utilizes the useful fact that the optimal local bandwidth depends on f''(x). The main idea is to first partition the domain of f(x) into different segments for which the second derivative of each segment is approximately constant. The number and the length of the segments are assumed unknown and will be estimated. Then, after such a partition is obtained, any reliable, well-studied global bandwidth selection method can be applied to choose the bandwidth for each segment. The empirical performance of the proposed local bandwidth selection method is evaluated by numerical experiments.