The Annals of Statistics

Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra

Michael H. Neumann and Rainer von Sachs

Full-text: Open access

Abstract

We derive minimax rates for estimation in anisotropic smoothness classes. These rates are attained by a coordinatewise thresholded wavelet estimator based on a tensor product basis with separate scale parameter for every dimension. It is shown that this basis is superior to its one-scale multiresolution analog, if different degrees of smoothness in different directions are present.

As an important application we introduce a new adaptive waveletestimator of the time-dependent spectrum of a locally stationary time series. Using this model which was recently developed by Dahlhaus, we show that the resulting estimator attains nearly the rate, which is optimal in Gaussian white noise, simultaneously over a wide range of smoothness classes. Moreover, by our new approach we overcome the difficulty of how to choose the right amount of smoothing, that is, how to adapt to the appropriate resolution, for reconstructing the local structure of the evolutionary spectrum in the time-frequency plane.

Article information

Source
Ann. Statist., Volume 25, Number 1 (1997), 38-76.

Dates
First available in Project Euclid: 10 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1034276621

Digital Object Identifier
doi:10.1214/aos/1034276621

Mathematical Reviews number (MathSciNet)
MR1429917

Zentralblatt MATH identifier
0871.62081

Subjects
Primary: 62G07: Density estimation 62M15: Spectral analysis
Secondary: 62E20: Asymptotic distribution theory 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Keywords
Anisotropic smoothness classes adaptive estimation optimal rate of convergence wavelet thresholding tensor product basis time-frequency plane locally stationary time series evolutionary spectrum

Citation

Neumann, Michael H.; von Sachs, Rainer. Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra. Ann. Statist. 25 (1997), no. 1, 38--76. doi:10.1214/aos/1034276621. https://projecteuclid.org/euclid.aos/1034276621


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References

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