Optimal change-point estimation from indirect observations

The Annals of Statistics · 2006 · Volume 34 · Issue 1

Abstract

We study nonparametric change-point estimation from indirect noisy observations. Focusing on the white noise convolution model, we consider two classes of functions that are smooth apart from the change-point. We establish lower bounds on the minimax risk in estimating the change-point and develop rate optimal estimation procedures. The results demonstrate that the best achievable rates of convergence are determined both by smoothness of the function away from the change-point and by the degree of ill-posedness of the convolution operator. Optimality is obtained by introducing a new technique that involves, as a key element, detection of zero crossings of an estimate of the properly smoothed second derivative of the underlying function.

Keywords

62G05, 62G20, Change-point estimation, Deconvolution, ill-posedness, minimax risk, Optimal rates of convergence, probe functional

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