Wavelet Thresholding for Non (Necessarily) Gaussian Noise

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

Author dc:creator

• Averkamp, Roland

Contributors dc:contributor

• Rüschendorf, Ludger

• wavelet
• nonparametric curve estimation
• oracle inequality
• minimax

Repository record source_url

https://freidok.uni-freiburg.de/data/10

OAI identifier oai:identifier

oai:freidok.uni-freiburg.de:10