Minimax estimation of sharp change points

Abstract

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.