Some statistical results in high-dimensional dependence modeling

This thesis can be divided into three parts. In the first part, we study adaptivity to the noise level in the high-dimensional linear regression framework. We prove that two square-root estimators attains the minimax rates of estimation and prediction. We show that a corresponding median-of-means version can still attains the same optimal rates while being robust to outliers in the data.

The second part is devoted to the analysis of several conditional dependence models. We propose some tests of the simplifying assumption that a conditional copula is constant with respect to its conditioning event, and prove the consistency of a semiparametric bootstrap scheme.
If the conditional copula is not constant with respect to the conditional event, then it can be modelled using the corresponding Kendall's tau. We study the estimation of this conditional dependence parameter using 3 different approaches : kernel techniques, regression-type models and classification algorithms.

The last part regroups two different topics in inference. We review and propose estimators for regular conditional functionals using U-statistics. Finally, we study the construction and the theoretical properties of confidence intervals for ratios of means under different sets of assumptions and paradigms.