Wavelet Thresholding: Beyond the Gaussian I.I.D. Situation

  • With this article we first like to give a brief review on wavelet thresholding methods in non-Gaussian and non-i.i.d. situations, respectively. Many of these applications are based on Gaussian approximations of the empirical coefficients. For regression and density estimation with independent observations, we establish joint asymptotic normality of the empirical coefficients by means of strong approximations. Then we describe how one can prove asymptotic normality under mixing conditions on the observations by cumulant techniques.; In the second part, we apply these non-linear adaptive shrinking schemes to spectral estimation problems for both a stationary and a non-stationary time series setup. For the latter one, in a model of Dahlhaus on the evolutionary spectrum of a locally stationary time series, we present two different approaches. Moreover, we show that in classes of anisotropic function spaces an appropriately chosen wavelet basis automatically adapts to possibly different degrees of regularity for the different directions. The resulting fully-adaptive spectral estimator attains the rate that is optimal in the idealized Gaussian white noise model up to a logarithmic factor.

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Author:Michael H. Neumann, Rainer von Sachs
URN (permanent link):urn:nbn:de:hbz:386-kluedo-5338
Serie (Series number):Berichte der Arbeitsgruppe Technomathematik (AGTM Report) (133)
Document Type:Article
Language of publication:English
Year of Completion:1995
Year of Publication:1995
Publishing Institute:Technische Universität Kaiserslautern
Date of the Publication (Server):2000/04/03
Tag:Non-linear wavelet thresholding; evolutionary spectrum; non-Gaussia non-i.i.d. errors; nonparametric regression and (spectral) density estimation
Source:Wavelets and Statistics (A. Antoniadis, G. Oppenheim, eds.) Springer LN Statistics 103, 301-329 (1995)
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
Licence (German):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011