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.