Periodic boxcar deconvolution and diophantine approximation

Details Periodic boxcar deconvolution and diophantine approximation Periodic boxcar deconvolution and diophantine approximation

2005-03-29

arxiv.org/abs/math/0503663v1

Annals of Statistics 2004, Vol. 32, No. 5, 1781-1804

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053604000000391 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

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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