We consider in this paper the statistical linear inverse problem Y = Af + ϵξ where A denotes a compact operator, ϵ a noise level and ξ a stochastic noise. The unknown function f has to be recovered from the indirect measurement Y. We are interested in the following approach: given a family of estimators, we want to select the best possible one. In this context, the unbiased risk estimation (URE) method is rather popular. Nevertheless, it is also very unstable. Recently, Cavalier and Golubev (2006) introduced the risk hull minimization (RHM) method. It significantly improves the performances of the standard URE procedure. However, it only concerns projection rules. Using recent developments on ordered processes, we prove in this paper that it can be extended to a large class of linear estimators.
Mots-clés : inverse problems, oracle inequality, ordered process, risk hull and Tikhonov estimation
@article{PS_2010__14__409_0,
author = {Marteau, Cl\'ement},
title = {Risk hull method for spectral regularization in linear statistical inverse problems},
journal = {ESAIM: Probability and Statistics},
pages = {409--434},
publisher = {EDP-Sciences},
volume = {14},
year = {2010},
doi = {10.1051/ps/2009011},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2009011/}
}
TY - JOUR AU - Marteau, Clément TI - Risk hull method for spectral regularization in linear statistical inverse problems JO - ESAIM: Probability and Statistics PY - 2010 SP - 409 EP - 434 VL - 14 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2009011/ DO - 10.1051/ps/2009011 LA - en ID - PS_2010__14__409_0 ER -
%0 Journal Article %A Marteau, Clément %T Risk hull method for spectral regularization in linear statistical inverse problems %J ESAIM: Probability and Statistics %D 2010 %P 409-434 %V 14 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2009011/ %R 10.1051/ps/2009011 %G en %F PS_2010__14__409_0
Marteau, Clément. Risk hull method for spectral regularization in linear statistical inverse problems. ESAIM: Probability and Statistics, Tome 14 (2010), pp. 409-434. doi : 10.1051/ps/2009011. http://archive.numdam.org/articles/10.1051/ps/2009011/
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