# Statistical Estimation in High Dimension, Sparsity and Oracle Inequalities

Abstract : We treat two subjects. The first subject is about statistical learning in high-dimension, that is when the number of paramaters to estimate is larger than the sample size. In this context, the generally adopted assumption is that the number of true parameters is much smaller than the number of potential paramaters. This assumption is called the \emph{sparsity assumption}''. We study the statistical properties of two types of procedures: the penalized risk minimization procedures with a $l_{1}$ penalty term on the set of potential parameters and the exponential weights procedures. The second subject is about the study of two aggregation procedures in a density estimation problem. We establish oracle inequalities for the $L^{\pi}$ norm, $1\leqslant \pi \leqslant \infty$. Next, we exploit these results to build minimax rate adaptive estimators of the density.
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https://tel.archives-ouvertes.fr/tel-00435917
Contributor : Karim Lounici <>
Submitted on : Wednesday, November 25, 2009 - 12:09:38 PM
Last modification on : Tuesday, May 14, 2019 - 11:08:25 AM
Long-term archiving on : Thursday, June 17, 2010 - 9:59:51 PM

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• HAL Id : tel-00435917, version 1

### Citation

Karim Lounici. Statistical Estimation in High Dimension, Sparsity and Oracle Inequalities. Mathematics [math]. Université Paris-Diderot - Paris VII, 2009. English. ⟨tel-00435917⟩

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