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October 1999 Speed of convergence for the blind deconvolution of a linear system with discrete random input
E. Gassiat, E. Gautherat
Ann. Statist. 27(5): 1684-1705 (October 1999). DOI: 10.1214/aos/1017939147

Abstract

In a recent paper, we proposed a new estimation method for the blind deconvolution of a linear system with discrete random input, when the observations may be noise perturbed. We give here asymptotic properties of the estimators in the parametric situation. With nonnoisy observations, the speed of convergence is governed by the $l_1$-tail of the inverse filter, which may have an exponential decrease. With noisy observations, the estimator satisfies a limit theorem with known distribution, which allows for the construction of confidence regions. To our knowledge, this is the first precise asymptotic result in the noisy blind deconvolution problem with an unknown level of noise. We also extend results concerning Hankel’s estimation to Toeplitz’s estimation and prove a formula to compute Toeplitz forms that may have interest in itself.

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E. Gassiat. E. Gautherat. "Speed of convergence for the blind deconvolution of a linear system with discrete random input." Ann. Statist. 27 (5) 1684 - 1705, October 1999. https://doi.org/10.1214/aos/1017939147

Information

Published: October 1999
First available in Project Euclid: 23 September 2004

zbMATH: 0965.62070
MathSciNet: MR2001D:62088
Digital Object Identifier: 10.1214/aos/1017939147

Subjects:
Primary: 62G05
Secondary: 62M09

Rights: Copyright © 1999 Institute of Mathematical Statistics

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Vol.27 • No. 5 • October 1999
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