Generalized Functional Partially Linear Single-index Models

Abstract

Single-index models are potentially important tools for multivariate nonparametric regression analysis. They generalize linear regression models by replacing the linear combination \(\alpha^T_0\) with a nonparametric component \(\eta_0({\alpha^T_0})X\), where \(\eta_0(\cdot)\) is an unknown univariate link function. [7] studied generalized partially linear singleindex models (GPLSIM) where the systematic component of the model has a flexible semi-parametric form with a general link function. In this paper, we generalize these models to have a functional component, replacing the generalized partially linear single-index models \(\eta_0({\alpha^T_0}) + \beta^T_0 Z {\text by} \eta_0({\alpha^T_0}){\int^1_0} \beta^T_0 (t)Z(t) dt\), where α is a vector in \(\mathbb{R}^d\), \(\eta_0(\cdot) \text and \beta_0(\cdot)\) are unknown functions which are to be estimated. We propose estimators of the unknown parameter α0 and the unknown functions \(\beta_0(\cdot) \,{\text and} \,\eta_0 (\cdot)\) and we establish their asymptotic distributions. Then, we illustrate through some examples the models and the effectiveness of the proposed estimation methodology.