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URN: urn:nbn:de:bsz:25-opus-102
URL: http://www.freidok.uni-freiburg.de/volltexte/10/

Averkamp, Roland

Wavelet Thresholding for Non (Necessarily) Gaussian Noise

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Kurzfassung in Deutsch

Soon after the discovery of orthonormal wavelets, in particular the
compactly supported ones,
these wavelets have been used for non-parametric function estimation.
In the literature, two main models are present. In the first model,
the target functions are members of some smoothness class and for this class
the minimax properties of estimators are investigated. This approach is
facilitated by the fact that some smoothness spaces can naturally be
described by norms of sequences of wavelet coefficients.
In the second approach the risk of an estimator is compared to the risk of
an ``ideal'' estimator. This ``estimator'' is ``ideal'' because it has some
knowledge of the wavelet coefficients of the function to estimate, so it is
not really an estimator.
The quality of estimation is then measured by the size of the ratio of the estimators risk and the risk of the ideal estimator.
First both models have been mainly studied for Gaussian noise.
Later the first model was investigated by others for other types of noises.
The second model was investigated by Gao for non-Gaussian noise.
In this thesis I will consider both types of approaches for non-Gaussian noise.
The content of this thesis is as follows:
In the first two chapters I give a short introduction to wavelets and their
use in non-parametric function estimation.
The third chapter is about the ideal estimator approach for non-Gaussian noise.
The fourth chapter deals with the function space approach: an addition to
known results is obtained and the performance of wavelet thresholding for
median filtered data is investigated.
The subject of chapter 5 is an extension of Stein's unbiased risk estimation
for general classes of infinitely divisible noise in the location model.
Stein's unbiased risk estimate is the basis for a very
adaptive thresholding estimator.
The last chapter presents a comparison of the thresholds in the two
approaches and a connection to kernel estimators.

SWD-Schlagwörter: Wavelet , Minimax-Schätzung , Nichtparametrische Schätzung
Freie Schlagwörter (englisch): wavelet , nonparametric curve estimation , oracle inequality , minimax
MSC Klassifikation 60g70 , 62g07 , 41a25 , 60g70 , 62g07 , 41a25
Institut: Institut für Mathematische Stochastik
Fakultät: Mathematische Fakultät (bis Sept. 2002)
DDC-Sachgruppe: Mathematik
Dokumentart: Dissertation
Erstgutachter: Prof. Dr. L. Rüschendorf
Sprache: Englisch
Tag der mündlichen Prüfung: 27.10.1999
Erstellungsjahr: 1999
Publikationsdatum: 16.12.1999