Regression curve estimation with possible jumps/roofs/valleys preserved and step/roof edge detection in regression surface

Curve estimation from observed data with noise has broad applications. In certain applications, the underlying regression curve has jumps or roofs/valleys at some unknown positions, representing structural changes of the related process. Detection of such changes is therefore important for understanding the structural changes and for estimating the regression curve properly. 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 dissertation, 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. Generally speaking, first order derivatives of the true regression curve are sensitive to jumps, but they can not localize the detected jumps well. As a comparison, second order derivatives can localize the detected jumps well, although they are often sensitive to noise. Our proposed jump detection procedure tries to make use of helpful information about jumps in both the first order and the second order derivatives of the true regression curve. Roofs or valleys in the true regression curve can be regarded as jumps in the first order derivative of the regression curve. Similar to jump detection, our proposed roof/valley detection procedure is based on the second and third order derivatives of the true regression curve. Using detected jumps and roofs/valleys, a curve estimation procedure is also proposed, which is able to preserve jumps and roofs/valleys when removing noise. Theoretical justifications and numerical studies show that they work well in applications. In this dissertation, the proposed 1-dimensional (1-D) jump/roof/valley detection procedures are also extended to 2-dimensional (2-D) cases, where jump/roof/valley detection in regression surfaces is of interest. Numerical studies show that 2-D such procedures perform well.