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2011 Penalized wavelets: Embedding wavelets into semiparametric regression
M.P. Wand, J.T. Ormerod
Electron. J. Statist. 5: 1654-1717 (2011). DOI: 10.1214/11-EJS652

Abstract

We introduce the concept of penalized wavelets to facilitate seamless embedding of wavelets into semiparametric regression models. In particular, we show that penalized wavelets are analogous to penalized splines; the latter being the established approach to function estimation in semiparametric regression. They differ only in the type of penalization that is appropriate. This fact is not borne out by the existing wavelet literature, where the regression modelling and fitting issues are overshadowed by computational issues such as efficiency gains afforded by the Discrete Wavelet Transform and partially obscured by a tendency to work in the wavelet coefficient space. With penalized wavelet structure in place, we then show that fitting and inference can be achieved via the same general approaches used for penalized splines: penalized least squares, maximum likelihood and best prediction within a frequentist mixed model framework, and Markov chain Monte Carlo and mean field variational Bayes within a Bayesian framework. Penalized wavelets are also shown have a close relationship with wide data (“pn”) regression and benefit from ongoing research on that topic.

Citation

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M.P. Wand. J.T. Ormerod. "Penalized wavelets: Embedding wavelets into semiparametric regression." Electron. J. Statist. 5 1654 - 1717, 2011. https://doi.org/10.1214/11-EJS652

Information

Published: 2011
First available in Project Euclid: 13 December 2011

zbMATH: 1271.62089
MathSciNet: MR2870147
Digital Object Identifier: 10.1214/11-EJS652

Rights: Copyright © 2011 The Institute of Mathematical Statistics and the Bernoulli Society

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