La Convergence Presque Complète En Statistique Semi-paramétrique Fonctionnelle

Résumé: This thesis explores several estimation methods for functional data within the framework of functional regression models, with an emphasis on kernel-based techniques. Our contribution lies in the reformulation of the estimator proposed by Ferraty and Vieu [23], allowing us to estimate the distribution function of the response variable using a single kernel. This kernel focuses on the functional part, enabling the simulation of the conditional cumulative distribution function derived from the regression function, and its comparison with the theoretical distribution function. In the first part, we examine the estimation of the regression operator in a single-index functional model using kernel methods. We apply several techniques, including Leave-One-Out (LOO) cross-validation, to select the optimal smoothing parameter. The results demonstrate the almost complete convergence of the kernel estimator, validating its theoretical foundations and practical efficacy. The second part focuses on the estimation of the conditional cumulative distribution function for functional data. We use a bandwidth selection approach and validate the results by studying the almost complete convergence of the conditional cumulative distribution estimator. Through simulations, we confirm the robustness and predictive capabilities of the proposed kernel method, while emphasizing the importance of choosing appropriate smoothing parameters. In the third part, we address the challenges associated with managing two smoothing parameters in the double-kernel estimators used for conditional distribution estimation. By reformulating the approach, we demonstrate the practical benefits of using a single kernel, simplifying the estimation process. We recommend the use of cross-validation to optimize the selection of these parameters, further improving the flexibility and efficiency of the estimator for prediction.

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