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October 2004 Periodic boxcar deconvolution and diophantine approximation
Iain M. Johnstone, Marc Raimondo
Ann. Statist. 32(5): 1781-1804 (October 2004). DOI: 10.1214/009053604000000391


We consider the nonparametric estimation of a periodic function that is observed in additive Gaussian white noise after convolution with a “boxcar,” the indicator function of an interval. This is an idealized model for the problem of recovery of noisy signals and images observed with “motion blur.” If the length of the boxcar is rational, then certain frequencies are irretreviably lost in the periodic model. We consider the rate of convergence of estimators when the length of the boxcar is irrational, using classical results on approximation of irrationals by continued fractions. A basic question of interest is whether the minimax rate of convergence is slower than for nonperiodic problems with 1/f-like convolution filters. The answer turns out to depend on the type and smoothness of functions being estimated in a manner not seen with “homogeneous” filters.


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Iain M. Johnstone. Marc Raimondo. "Periodic boxcar deconvolution and diophantine approximation." Ann. Statist. 32 (5) 1781 - 1804, October 2004.


Published: October 2004
First available in Project Euclid: 27 October 2004

zbMATH: 1056.62044
MathSciNet: MR2102493
Digital Object Identifier: 10.1214/009053604000000391

Primary: 62G20
Secondary: 11K60 , 65R32

Keywords: Continued fraction , Deconvolution , ellipsoid , hyperrectangle , Ill-posed problem , irrational number , Linear inverse problem , minimax risk , motion blur , nonparametric estimation , rates of convergence

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 5 • October 2004
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