Electronic Journal of Statistics

Data-driven wavelet-Fisz methodology for nonparametric function estimation

Piotr Fryzlewicz

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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.

Article information

Electron. J. Statist., Volume 2 (2008), 863-896.

First available in Project Euclid: 1 October 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression
Secondary: 62G05: Estimation 62G20: Asymptotic properties

Besov spaces exponential inequality heteroscedasticity Nadaraya-Watson estimator nonparametric regression variance function variance-stabilising transform wavelets


Fryzlewicz, Piotr. Data-driven wavelet-Fisz methodology for nonparametric function estimation. Electron. J. Statist. 2 (2008), 863--896. doi:10.1214/07-EJS139. https://projecteuclid.org/euclid.ejs/1222868026

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