Risk hull method for spectral regularization in linear statistical inverse problems

Clément Marteau (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

We consider in this paper the statistical linear inverse problem $Y=Af+\epsilon\xi$ where $A$ denotes a compact operator, $\epsilon$ a noise level and $\xi$ 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

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Clément Marteau. Risk hull method for spectral regularization in linear statistical inverse problems. ESAIM: Probability and Statistics, 2010, 14, pp.409-434. ⟨hal-00369284⟩