Optimal change-point estimation in inverse problems

We develop a method of estimating change{points of a function in the case of indirect noisy observations. As two paradigmatic problems we consider de-convolution and errors-in-variables regression. We estimate the scalar products of our indirectly observed function with appropriate test functions, which are shifted over the interval of interest. An estimator of the change point is obtained by the extremal point of this quantity. We derive rates of convergence for this estimator. They depend on the degree of ill-posedness of the problem, which derives from the smoothness of the error density. Analyzing the Hellinger modulus of continuity of the problem we show that these rates are minimax. 1 1. Introduction Change-point estimation has often been studied in the regression context. There are many practical motivations why one is interested in knowing such points of rapid change. Sometimes there is real scientiic interest in these point, but one can also exploit knowledge about them for estimation purposes themselves. The simplest case is that of a single jump of an otherwise smooth function. The optimal rate at which then a change-point can be estimated is known to be n ?1. Korostelev (1987) derives an optimal method in the Gaussian white noise model, which can also be applied in the usual nonparametric regression setting. Another popular approach is based on the analysis of diierences of certain kernel estimators; see, e. considers a closely related method based on wavelets. It can be shown that one can achieve the optimal rate of convergence also by the kernel-based method, provided one uses an appropriate, necessarily discontinuous kernel. In the present paper we study this problem in the context of ill-posed inverse problems. Such problems arise when we can observe an object of interest only indirectly. Typical settings are deconvolution, errors-in-variables regression, estimation of mixing densities, image blur models and image reconstruction in computerized tomography. The quality at which a function can be estimated from such indirect, noisy observations depends on the degree of ill-posedness of the problem. For example , deconvolution becomes harder as smoothness of the error distribution increases. Most of the available results focus on the estimation of functions with homogeneous However, in practical applications one is often confronted with functions that have quite inhomogeneous smoothness characteristics: they are quite smooth on one part of the domain, but much less regular on another part. In such situations usual linear smoothing methods, which apply a …