Kernel Methods for Optimal Change-Points Estimation in Derivatives

In this article we propose an implementation of the so-called zero-crossing-time detection technique specifically designed for estimating the location of jump-points in the first derivative (kinks) of a regression function f . Our algorithm relies on a new class of kernel functions having a second derivative with vanishing moments and an asymmetric first derivative steep enough near the origin. We provide a software package which, for a sample of size n, produces estimators with an accuracy of order, at least, O(n−2/5). This contrasts with current algorithms for kink estimation which at best provide an accuracy of order O(n−1/3). In the software, the kernel statistic is standardized and compared to the universal threshold to test the existence of a kink. A simulation study shows that our algorithm enjoys very good finite sample properties even for low sample sizes. The method reveals kink features in real datasets with high noise levels at places where traditional smoothers tend to oversmooth the data.