Non-parametric estimation of the density of hidden random variables.

Summary This thesis includes several non-parametric probability density estimation procedures.In each case the variables of interest are not directly observed, which is a major difficulty.The first part deals with a linear mixed model where repeated observations are available.The second part is concerned with stochastic differential equation models with random effects.Several trajectories are observed in continuous time over a common time interval.The third part is placed in a multiplicative noise context.The different parts of this thesis are connected by a common inverse problem context and by a discussion of a common problem. The third part is placed in a context of multiplicative noise.The different parts of this thesis are linked by a common context of inverse problem and by a common problem: the estimation of the density of a hidden variable. In the first two parts the density of one or more random effects is estimated. In the third part, the density of the original variable is reconstructed from noisy observations. Different global estimation methods are used to build efficient estimators: kernel estimators, projection estimators or estimators built by deconvolution. The selection of parameters leads to adaptive estimators and the integrated quadratic risks are increased thanks to a Talagrand concentration inequality. A simulation study of each estimator illustrates their performance. A neural dataset is studied thanks to the procedures set up for stochastic differential equations.