Wavelet Thresholding in Anisotropic Function Classes and Application to Adaptive Estimation of Evolutionary Spectra

  • We derive minimax rates for estimation in anisotropic smoothness classes. This rate is 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 wavelet estimator of the time-dependent spectrum of a locally stationary time series. Using this model which was resently 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, i.e. how to adapt to the appropriate resolution, for reconstructing the local structure of the evolutionary spectrum in the time-frequency plane.

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Author:Michael H. Neumann, Rainer von Sachs
URN (permanent link):urn:nbn:de:hbz:386-kluedo-5327
Serie (Series number):Berichte der Arbeitsgruppe Technomathematik (AGTM Report) (132)
Document Type:Article
Language of publication:English
Year of Completion:1997
Year of Publication:1997
Publishing Institute:Technische Universität Kaiserslautern
Date of the Publication (Server):2000/04/03
Tag:Anisotropic smoothness classes; adaptive estimation; optimal rate of convergence; tensor product basis; time-frequency plan; wavelet thresholding
Source:Annals of Statistics, 25, 38-76 (1997)
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):62-XX STATISTICS / 62Exx Distribution theory [See also 60Exx] / 62E20 Asymptotic distribution theory
62-XX STATISTICS / 62Gxx Nonparametric inference / 62G07 Density estimation
62-XX STATISTICS / 62Mxx Inference from stochastic processes / 62M10 Time series, auto-correlation, regression, etc. [See also 91B84]
62-XX STATISTICS / 62Mxx Inference from stochastic processes / 62M15 Spectral analysis
Licence (German):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011