[en] This doctoral thesis is dedicated to functional regression for scalar response. In particular, we focus on functional semiparametric models, which combine the practical advantages of parametric and nonparametric approaches, surpassing both methodologies. Accordingly, several semiparametric models involving a functional singleindex component were studied from a theoretical and practical perspective. First, for the functional single-index model (FSIM) and the semi-functional partial linear single-index model (SFPLSIM), we provide uniform consistency results (over all parameters involved) for kernel- and k-Nearest-Neighbours-based statistics related to the estimation of the semiparametric component. Second, for the sparse semifunctional partial linear single-index model (SSFPLSIM), we develop a variable selection procedure in the linear component based on penalized least squares (PLS). The good behaviour of this method is theoretically assured (rates of convergence of the estimators are obtained, as well as asymptotic behaviour of the variable selection procedure). Third, the SSFPLSIM is adapted to the case in which covariates with linear e ect come from the discretization of a curve. For this new model, the multi-functional partial linear single-index model (MFPLSIM), the variable selection problem was also studied. Consequently, two new algorithms were proposed (providing theoretical results that ensure their good performance) to solve the ine ciency of the PLS method when it is directly applied to the MFPLSIM. For all the models and procedures mentioned above, theoretical results are accompanied by both simulation studies and real data applications which illustrate the good performance of the proposed methodology in practice.

[gl] Esta tese está adicada ao estudo da regresión funcional con variable resposta escalar. En particular, centrámonos en modelos funcionais semi-paramétricos, os cales combinan as vantaxes prácticas dos enfoques paramétrico e non-paramétrico, superando a ambas metodoloxías. Desta maneira, estudáronse, tanto dende o punto de vista te orico como dende a perspectiva pr actica, varios modelos semi-param etricos que involucran unha compoñente funcional single-index. En primeiro lugar, para o functional single-index model (FSIM) e para o semi-functional partial linear single-index model (SFPLSIM) establecemos resultados de consistencia uniforme (sobre todos os parámetros involucrados) para os estatísticos de tipo núcleo e tipo k-veciños-máis-próximos relacionados coa estimación da compoñente semi-paramétrica do modelo. En segundo lugar, para o sparse semi-functional partial linear single-index model (SSFPLSIM) desenvolvemos un procedemento de selección de variables na compoñente linear baseado en mínimos cadrados penalizados (PLS, iniciais de penalized least squares). O bo comportamento deste método asegurouse dende o punto de vista teórico (obtendo taxas de converxencia dos estimadores, así como o comportamento asintótico do procedemento de selección de variables). En terceiro lugar, o SSFPLSIM adaptouse ao escenario no cal as covariables con efecto linear proveñen da discretización dunha curva. Para este novo modelo, o multi-functional partial linear single-index model (MFPLSIM), estudouse tamén o problema da selección de variables e propuxéronse dous novos algoritmos (dos que aseguramos teoricamente o seu bo comportamento) para resolver a inefi cacia do método PLS cando se aplica directamente ao MFPLSIM. Para todos os modelos e procedementos citados, os resultados teóricos acompañáronse de estudos de simulación e aplicacións a datos reais que ilustran o bo comportamento na práctica da metodoloxía presentada.