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Biometrika 2006 93(2):465-471; doi:10.1093/biomet/93.2.465
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© 2006 Biometrika Trust

Miscellanea

Parametric modelling of thresholds across scales in wavelet regression

Anestis Antoniadis1 and Piotr Fryzlewicz2

1 Laboratoire de Modelisation et Calcul, Université Joseph Fourier, Tour IRMA, B.P.53, 38041 Grenoble CEDEX 9, France. anestis.antoniadis{at}imag.fr, 2 Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, U.K. p.z.fryzlewicz{at}bristol.ac.uk

We propose a parametric wavelet thresholding procedure for estimation in the ‘function plus independent, identically distributed Gaussian noise’ model. To reflect the decreasing sparsity of wavelet coefficients from finer to coarser scales, our thresholds also decrease. They retain the noise-free reconstruction property while being lower than the universal threshold, and are jointly parameterised by a single scalar parameter. We show that our estimator achieves near-optimal risk rates for the usual range of Besov spaces. We propose a crossvalidation technique for choosing the parameter of our procedure. A simulation study demonstrates very good performance of our estimator compared to other state-of-the-art techniques. We discuss an extension to non-Gaussian noise.

Key Words: Asymptotic rate; Besov space; Noise-free reconstruction; Thresholding; Wavelet decomposition.

Received October 2004. Revised October 2005.


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