Abstract
Single-index models are potentially important tools for multivariate non-parametric regression analysis. They generalize linear regression models by replacing the linear combination α0⊤X with a non-parametric component η0α0⊤X, where η0(·) 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 η0α0⊤X+β0⊤Z, where α is a vector in IRd, η0(·) and β0(·) are unknown functions that are to be estimated. We propose estimates of the unknown parameter α0, the unknown functions β0(·) and η0(·) 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.
Highlights
IntroductionGLMs assume the responses come from the exponential dispersion model family
Generalized linear models are proposed by Nelder and Wedderburn [1], g(μ(X)) = β> X; for a detail review, we refer the readers to McCullagh and Nelder [2]; it consists of a random component and systematic component
We introduce estimates for the Generalized Partially Linear Single-Index
Summary
GLMs assume the responses come from the exponential dispersion model family. They extend linear models to allow the relationship between the predictors and the function of the mean of continuous or discrete response through a canonical link function. These models encounter problems such as the canonical link function is sometimes unknown, the link between response and predictors can be complex as well as the plague of dimension reduction. To address these problems, several approaches have been developed. The manuscripts of Wood [4] and Dunn, Peter, Smyth, Gordon [5] are the latest references dealing with these two models
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