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
paper, 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. Theoretical justifications and numerical
studies show that they work well in applications.