The Annals of Statistics

Periodic boxcar deconvolution and diophantine approximation

Iain M. Johnstone and Marc Raimondo

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Abstract

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.

Article information

Source
Ann. Statist., Volume 32, Number 5 (2004), 1781-1804.

Dates
First available in Project Euclid: 27 October 2004

Permanent link to this document
https://projecteuclid.org/euclid.aos/1098883772

Digital Object Identifier
doi:10.1214/009053604000000391

Mathematical Reviews number (MathSciNet)
MR2102493

Zentralblatt MATH identifier
1056.62044

Subjects
Primary: 62G20: Asymptotic properties
Secondary: 65R32: Inverse problems 11K60: Diophantine approximation [See also 11Jxx]

Keywords
Continued fraction deconvolution ellipsoid hyperrectangle ill-posed problem irrational number linear inverse problem minimax risk motion blur nonparametric estimation rates of convergence

Citation

Johnstone, Iain M.; Raimondo, Marc. Periodic boxcar deconvolution and diophantine approximation. Ann. Statist. 32 (2004), no. 5, 1781--1804. doi:10.1214/009053604000000391. https://projecteuclid.org/euclid.aos/1098883772


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