Contribution to the statistical analysis of functional data.

Summary In this thesis, we focus on functional data. The generalization of the generalized linear functional model to the model defined by estimating equations is studied. We obtain a central limit theorem for the considered estimator. The optimal instruments are estimated, and we obtain a uniform convergence of the estimators. We are then interested in different tests in functional data. These are non-parametric tests to study the effect of a functional random covariate on an error term, which can be directly observed as a response or estimated from a functional model like the functional linear model. In order to implement the different tests, we have proven a dimension reduction result that relies on projections of the functional covariate. We construct non-effect and goodness-of-fit tests using either kernel smoothing or nearest neighbor smoothing. A goodness-of-fit test in the functional linear model is proposed. All these tests are studied from a theoretical and practical point of view.