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2008 Data-driven wavelet-Fisz methodology for nonparametric function estimation
Piotr Fryzlewicz
Electron. J. Statist. 2: 863-896 (2008). DOI: 10.1214/07-EJS139


We propose a wavelet-based technique for the nonparametric estimation of functions contaminated with noise whose mean and variance are linked via a possibly unknown variance function. Our method, termed the data-driven wavelet-Fisz technique, consists of estimating the variance function via a Nadaraya-Watson estimator, and then performing a wavelet thresholding procedure which uses the estimated variance function and local means of the data to set the thresholds at a suitable level.

We demonstrate the mean-square near-optimality of our wavelet estimator over the usual range of Besov classes. To achieve this, we establish an exponential inequality for the Nadaraya-Watson variance function estimator.

We discuss various implementation issues concerning our wavelet estimator, and demonstrate its good practical performance. We also show how it leads to a new wavelet-domain data-driven variance-stabilising transform. Our estimator can be applied to a variety of problems, including the estimation of volatilities, spectral densities and Poisson intensities, as well as to a range of problems in which the distribution of the noise is unknown.


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Piotr Fryzlewicz. "Data-driven wavelet-Fisz methodology for nonparametric function estimation." Electron. J. Statist. 2 863 - 896, 2008.


Published: 2008
First available in Project Euclid: 1 October 2008

zbMATH: 1320.62090
MathSciNet: MR2447343
Digital Object Identifier: 10.1214/07-EJS139

Primary: 62G08
Secondary: 62G05 , 62G20

Keywords: Besov spaces , Exponential inequality , Heteroscedasticity , Nadaraya-Watson estimator , Nonparametric regression , variance function , variance-stabilising transform , Wavelets

Rights: Copyright © 2008 The Institute of Mathematical Statistics and the Bernoulli Society


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