Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T21:52:31.126Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundary coiflets for wavelet shrinkage in function estimation

Part of: Nonparametric inference Numerical methods in Fourier analysis Nontrigonometric harmonic 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: imj@stat.stanford.edu
Bernard W. Silverman
Affiliation:
St Peter's College, Oxford University, Oxford 0X1 2DL, UK. Email address: bemard.silverman@spc.ox.ac.uk
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Cohen, A., Daubechies, I. and Vial, P. (1993). Wavelets on the interval and fast wavelet transforms. Appl. Computational Harmonic Anal. 1, 54–81.CrossRefGoogle Scholar
[2] Cohen, A., Daubechies, I., Jawerth, B. and Vial, P. (1993). Multiresolution analysis, wavelets and fast algorithms on an interval. C. R. Acad. Sci. Paris 316, 417–421.Google Scholar
[3] Daubechies, I. (1993). Orthonormal bases of compactly supported wavelets. II. Variations on a theme. SIAM J. Math. Anal. 24, 499–519.CrossRefGoogle Scholar
[4] Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation via wavelet shrinkage. Biometrika 81, 425–455.CrossRefGoogle Scholar
[5] Donoho, D. L. and Johnstone, I. M. (1998). Minimax estimation via wavelet shrinkage. Ann. Statist. 26, 879–921.CrossRefGoogle Scholar
[6] Donoho, D. L. and Johnstone, I. M. (1999). Asymptotic minimaxity of wavelet estimators with sampled data. Statistica Sinica 9, 1–32.Google Scholar
[7] Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995). Wavelet shrinkage: asymptopia? J. R. Statist. Soc. B 57, 301–369.Google Scholar
[8] Johnstone, I. M. (2003). Function Estimation in Gaussian Noise: Sequence Models. Unpublished manuscript. Available at http://www-stat.stanford.edu/~imj/.Google Scholar
[9] Johnstone, I. M. and Silverman, B. W. (2003). Empirical Bayes selection of wavelet thresholds. Submitted. Available at http://www.stats.ox.ac.uk/~silverma/.Google Scholar
[10] Mallat, S. (1999). A Wavelet Tour of Signal Processing , 2nd edn. Academic Press, San Diego, CA.Google Scholar
[11] Okikiolu, G. O. (1971). Aspects of the Theory of Bounded Integral Operators in Lp-Spaces. Academic Press, London.Google Scholar
[12] Wojtaszczyk, P. (1997). A Mathematical Introduction to Wavelets. Cambridge University Press.CrossRefGoogle Scholar