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Estimation non-paramétrique de la densité de variables aléatoires cachées

Abstract : This thesis contains several nonparametric estimation procedures of a probability density function.In each case, the main difficulty lies in the fact that the variables of interest are not directly observed.The first part deals with a mixed linear model for which repeated observations are available.The second part focuses on stochastic differential equations with random effects. Many trajectories are observed continuously on the same time interval.The third part is in a full multiplicative noise framework.The parts of the thesis are connected by the same context of inverse problems and by a common problematic: the estimation of the density function of a hidden variable.In the first two parts the density of one or two random effects is estimated. In the third part the goal is to rebuild the density of the original variable from the noisy observations.Different global methods are used and lead to well competitive estimators: kernel estimators, projection estimators or estimators built from deconvolution.Parameter selection gives adaptive estimators and the integrated risks are bounded using a Talagrand concentration inequality.A simulation study for each proposed estimator highlights their performances.A neuronal dataset is investigated with the new procedures for stochastic differential equations developed in this work.
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  • HAL Id : tel-01685528, version 1



Charlotte Dion. Estimation non-paramétrique de la densité de variables aléatoires cachées. Variables complexes [math.CV]. Université Grenoble Alpes, 2016. Français. ⟨NNT : 2016GREAM031⟩. ⟨tel-01685528⟩



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