Jump Detection in a Regression Curve and Its Derivative

Curve estimation from observed noisy data has broad applications. In certain applications, the underlying regression curve may have singularities, including jumps and roofs/valleys (i.e., jumps in the first order derivative of the regression curve), at some unknown positions, representing structural changes of the related process. Detection of such singularities is important for understanding the structural changes. In the literature, a number of jump detection procedures have been proposed, most of which are based on estimation of the (one-sided) first order derivatives of the true regression curve. In this paper, motivated by certain related research in image processing, we propose an alternative jump detection procedure. Besides the first order derivatives, we suggest using helpful information about jumps in the second order derivatives as well. Theoretical justifications and numerical studies show that this jump detector works well in applications. This procedure is then extended for detecting roofs/valleys of the regression curve. Using detected jumps/roofs/valleys, a curve estimation procedure is also proposed, which can preserve possible jumps/roofs/valleys when removing noise. Note that the printed paper only provides outlines of proofs of several theorems in the paper. Complete proofs of the theorems are available online as supplemental materials.