Quadratic functional estimation in inverse problems

Abstract : We consider in this paper a Gaussian sequence model of observations $Y_i$, $i\geq 1$ having mean (or signal) $\theta_i$ and variance $\sigma_i$ which is growing polynomially like $i^\gamma$, $\gamma >0$. This model describes a large panel of inverse problems. We estimate the quadratic functional of the unknown signal $\sum_{i\geq 1}\theta_i^2$ when the signal belongs to ellipsoids of both finite smoothness functions (polynomial weights $i^\alpha$, $\alpha>0$) and infinite smoothness (exponential weights $e^{\beta i^r}$, $\beta >0$, $0\gamma+1/4$ or in the case of exponential weights), we obtain the parametric rate and the efficiency constant associated to it. Moreover, we give upper bounds of the second order term in the risk and conjecture that they are asymptotically sharp minimax. When the signal is finitely smooth with $\alpha \leq \gamma +1/4$, we compute non parametric upper bounds of the risk of and we presume also that the constant is asymptotically sharp.
Type de document :
Pré-publication, Document de travail
2009
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https://hal.archives-ouvertes.fr/hal-00361218
Contributeur : Katia Méziani <>
Soumis le : vendredi 13 février 2009 - 13:59:48
Dernière modification le : lundi 29 mai 2017 - 14:24:11
Document(s) archivé(s) le : mardi 8 juin 2010 - 18:23:53

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  • HAL Id : hal-00361218, version 1
  • ARXIV : 0902.2309

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PMA | INSMI | UPMC | USPC

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Cristina Butucea, Katia Méziani. Quadratic functional estimation in inverse problems. 2009. <hal-00361218>

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