Jump Detection in Regression Surfaces Using Both First-Order and Second-Order Derivatives

We consider the problem of detecting jump location curves of regression surfaces. In the literature, most existing methods detect jumps in regression surfaces based on estimation of either the first-order or the second-order derivatives of regression surfaces. Methods based on the first-order derivatives are usually better in removing noise, whereas methods based on the second-order derivatives are often superior in localization of the detected jumps. There are a number of existing jump detection procedures using both the first-order and second-order derivatives. But they often use the two types of derivatives in separate steps, and estimation of the derivatives is usually based on intuition. In this paper, we suggest a new procedure for jump detection in regression surfaces, which combines the two types of derivatives in a single criterion. Estimation of the derivatives is based on local quadratic kernel smoothing. Theoretical justifications and numerical studies show that it works well in applications.