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Boundary coiflets for wavelet shrinkage in function estimation

Part of: Nonparametric inference Nontrigonometric harmonic analysis Numerical methods in Fourier analysis

Published online by Cambridge University Press:  14 July 2016

Iain M. Johnstone  and
Bernard W. Silverman
Iain M. Johnstone
Affiliation:
Department of Statistics, Sequoia Hall, Stanford University, Stanford, CA 94305, USA. Email address: [email protected]
Bernard W. Silverman
Affiliation:
St Peter's College, Oxford University, Oxford 0X1 2DL, UK. Email address: [email protected]
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Abstract

There are standard modifications of certain compactly supported wavelets that yield orthonormal bases on a bounded interval. We extend one such construction to those wavelets, such as ‘coiflets', that may have fewer vanishing moments than had to be assumed previously. Our motivation lies in function estimation in statistics. We use these boundary-modified coiflets to show that the discrete wavelet transform of finite data from sampled regression models asymptotically provides a close approximation to the wavelet transform of the continuous Gaussian white noise model. In particular, estimation errors in the discrete setting of computational practice need not be essentially larger than those expected in the continuous setting of statistical theory.

Keywords

CoifletsGaussian white noise modelminimax riskmultiresolution analysisnonparametric regressionpreconditioningquadrature errorvanishing momentswavelet shrinkage

MSC classification

Primary: 62G08: Nonparametric regression 42C40: Wavelets and other special systems
Secondary: 62G20: Asymptotic properties 65T60: Wavelets
Type
Part 2. Estimation methods
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 81 - 98
Copyright
Copyright © Applied Probability Trust 2004 

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