Detecting abrupt changes by wavelet methods

The objective of this paper is to contribute to the methodology available for dealing with the detection and the estimation of the location of discontinuities in one-dimensional piecewise smooth regression functions observed in white Gaussian noise over an interval. Our approach is nonparametric in nature because the unknown function is not assumed to have any specific form. Our method relies upon a wavelet analysis of the observed signal and belongs to the class of "indirect" methods, where one detects and locates the change points prior to fitting the curve, and then uses ones favorite function estimation technique on each segment to recover the curve. We show that, provided discontinuities can be detected and located with sufficient accuracy, detection followed by wavelet smoothing enjoys optimal rates of convergence.