Some statistical results in high-dimensional dependence modeling.

Summary This thesis can be divided into three parts.In the first part, we study methods of adaptation to the noise level in the high dimensional linear regression model. We prove that two square-root estimators can reach the minimax estimation and prediction speeds. We show that a similar version built from mean medians can still reach the same optimal speeds and is robust to the presence of outliers.The second part is devoted to the analysis of several conditional dependence models. We propose several tests of the simplifying hypothesis that a conditional copula is constant with respect to its conditioning event, and we prove the consistency of a semi-parametric resampling technique. If the conditional copula is not constant with respect to its conditioning variable, then it can be modeled via its conditional Kendall's tau. We study the estimation of this conditional dependence parameter under 3 different approaches: kernel techniques, regression type models and classification algorithms.The last part gathers two contributions in the field of inference.We compare and propose different estimators of regular conditional functionals using U-statistics. Finally, we study the construction and theoretical properties of confidence intervals for mean ratios under different choices of assumptions and paradigms.