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August 2010 A deconvolution approach to estimation of a common shape in a shifted curves model
Jérémie Bigot, Sébastien Gadat
Ann. Statist. 38(4): 2422-2464 (August 2010). DOI: 10.1214/10-AOS800

Abstract

This paper considers the problem of adaptive estimation of a mean pattern in a randomly shifted curve model. We show that this problem can be transformed into a linear inverse problem, where the density of the random shifts plays the role of a convolution operator. An adaptive estimator of the mean pattern, based on wavelet thresholding is proposed. We study its consistency for the quadratic risk as the number of observed curves tends to infinity, and this estimator is shown to achieve a near-minimax rate of convergence over a large class of Besov balls. This rate depends both on the smoothness of the common shape of the curves and on the decay of the Fourier coefficients of the density of the random shifts. Hence, this paper makes a connection between mean pattern estimation and the statistical analysis of linear inverse problems, which is a new point of view on curve registration and image warping problems. We also provide a new method to estimate the unknown random shifts between curves. Some numerical experiments are given to illustrate the performances of our approach and to compare them with another algorithm existing in the literature.

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Jérémie Bigot. Sébastien Gadat. "A deconvolution approach to estimation of a common shape in a shifted curves model." Ann. Statist. 38 (4) 2422 - 2464, August 2010. https://doi.org/10.1214/10-AOS800

Information

Published: August 2010
First available in Project Euclid: 11 July 2010

zbMATH: 1202.62049
MathSciNet: MR2676894
Digital Object Identifier: 10.1214/10-AOS800

Subjects:
Primary: 62G08
Secondary: ‎42C40

Keywords: adaptive estimation , Besov space , Curve registration , Deconvolution , inverse problem , Mean pattern estimation , Meyer wavelets , Minimax rate

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 4 • August 2010
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