# 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.
Keywords :
Document type :
Preprints, Working Papers, ...
2009
Domain :

https://hal.archives-ouvertes.fr/hal-00361218
Contributor : Katia Méziani <>
Submitted on : Friday, February 13, 2009 - 1:59:48 PM
Last modification on : Wednesday, October 12, 2016 - 1:05:36 AM
Document(s) archivé(s) le : Tuesday, June 8, 2010 - 6:23:53 PM

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ButuceaMeziani.pdf
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### Identifiers

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

### Citation

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

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