Abstract
The main purpose of this thesis is to develop adaptive estimators for functional data. In the first part, we focus on the functional linear model and we propose a dimension selection device for projection estimators defined on both fixed and data-driven bases. The prediction error of the resulting estimators satisfies an oracle-type inequality and reaches the minimax rate of convergence. For the estimator defined on a data-driven approximation space, tools of perturbation theory are used to solve the problems related to the random nature of the collection of models. From a numerical point of view, this method of dimension selection is faster and more stable than the usual methods of cross validation. In a second part, we consider the problem of bandwidth selection for kernel estimators of the conditional cumulative distribution function when the covariate is functional. The method is inspired by the work of Goldenshluger and Lepski. The risk of the estimator is non-asymptotically upper-bounded. We also prove lower-bounds and establish that our estimator reaches the minimax convergence rate, up to an extra logarithmic term. In the last part, we propose an extension to a functional context of the response surface methodology, widely used in the industry. This work is motivated by an application to nuclear safety.