The Annals of Statistics

Periodic boxcar deconvolution and diophantine approximation

Iain M. Johnstone and Marc Raimondo

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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

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

First available in Project Euclid: 27 October 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

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