We define the sharp change point problem as an extension of earlier
problems in change point analysis related to nonparametric regression. As
particular cases, these include estimation of jump points in smooth curves.
More generally, we give a systematic treatment of the correct rate of
convergence for estimating the position of a “cusp”of an
arbitrary order. We propose a test function for the local regularity of a
signal that characterizes such a point as a global maximum. In the sample
implementation of our method, from observations of the signal at discrete time
positions $i/n, i =1 \ldots,n$, we use a wavelet transformation to approximate
the position of the change point in the no-noise case. We study the noise
effect, in the worst case scenario over a wide class of functions having a
unique irregularity of “order $\alpha$” and propose a sequence of
estimators which converge at the rate $n_{-1/(1+2\alpha)}$, as $n$ tends
to infinity. Finally we analyze the likelihood ration of the problem and show
that this is actually the minimaz rate of convertence. Examples of thresholding
empirical wavelet coefficients to estimate the position of sharp change points
are also presented.