Partially Linear Generalized Single Index Models for Functional Data (PLGSIMF)

Single-index models are potentially important tools for multivariate non-parametric regression analysis. They generalize linear regression models by replacing the linear combination α 0 ⊤ X with a non-parametric component η 0 α 0 ⊤ X , where η 0 ( · ) is an unknown univariate link function. In this article, we generalize these models to have a functional component, replacing the generalized partially linear single index models η 0 α 0 ⊤ X + β 0 ⊤ Z , where α is a vector in I R d , η 0 ( · ) and β 0 ( · ) are unknown functions that are to be estimated. We propose estimates of the unknown parameter α 0 , the unknown functions β 0 ( · ) and η 0 ( · ) and establish their asymptotic distributions, and furthermore, a simulation study is carried out to evaluate the models and the effectiveness of the proposed estimation methodology.