Methodological contributions in semiparametric regression models for functional data

Abstract

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 single-index 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 semi-functional 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 effect 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 inefficiency 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.