Cross-validation and penalization for density estimation.

Summary This thesis is based on the estimation of a density, considered from a non-parametric and non-asymptotic point of view. It deals with the problem of the selection of a kernel estimation method. The latter is a generalization of, among others, the problem of model selection and window selection. We study classical procedures, by penalization and resampling (in particular V-fold cross-validation), which evaluate the quality of a method by estimating its risk. We propose, thanks to concentration inequalities, a method to optimally calibrate the penalty to select a linear estimator and prove oracle inequalities and adaptation properties for these procedures. Moreover, a new resampled procedure, based on the comparison between estimators by robust tests, is proposed as an alternative to procedures based on the principle of unbiased risk estimation. A second objective is the comparison of all these procedures from a theoretical point of view and the analysis of the role of the V-parameter for the V-fold penalties. We validate the theoretical results by simulation studies.