High-Dimensional Statistics: Non-Parametric Generalized Functional Partially Linear Single-Index Model

Mohamed Alahiane, Idir Ouassou, Mustapha Rachdi and Philippe Vieu
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Mohamed Alahiane: Ecole Nationale des Sciences Appliquées, Université Cadi Ayyad, Marrakech 40000, Morocco
Idir Ouassou: Ecole Nationale des Sciences Appliquées, Université Cadi Ayyad, Marrakech 40000, Morocco
Mustapha Rachdi: Laboratoire AGEIS EA 7407, Université Grenoble Alpes, AGIM Team, UFR SHS, BP. 47, CEDEX 09, 38040 Grenoble, France
Philippe Vieu: Institut de Mathématiques de Toulouse, Université Paul Sabatier, CEDEX 09, 31062 Toulouse, France

Mathematics, 2022, vol. 10, issue 15, 1-21

Abstract: We study the non-parametric estimation of partially linear generalized single-index functional models, where the systematic component of the model has a flexible functional semi-parametric form with a general link function. We suggest an efficient and practical approach to estimate (I) the single-index link function, (II) the single-index coefficients as well as (III) the non-parametric functional component of the model. The estimation procedure is developed by applying quasi-likelihood, polynomial splines and kernel smoothings. We then derive the asymptotic properties, with rates, of the estimators of each component of the model. Their asymptotic normality is also established. By making use of the splines approximation and the Fisher scoring algorithm, we show that our approach has numerical advantages in terms of the practical efficiency and the computational stability. A computational study on data is provided to illustrate the good practical behavior of our methodology.

Keywords: functional data analysis; generalized linear model; polynomial splines; quasi-likelihood; semi-parametric regression; the kernel estimator of the regression operator; single-index model; Fisher scoring algorithm; asymptotic normality (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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