Generalized Partially Linear Single-Index Models

Abstract

The typical generalized linear model for a regression of a response Y on predictors (X, Z) has conditional mean function based on a linear combination of (X, Z). We generalize these models to have a nonparametric component, replacing the linear combination alpha(0)(T)X + beta(0)(T)Z by eta(0)(alpha(0)(T)X) + beta(0)(T)Z, where eta(0)(.) is an unknown function: We call these generalized partially lineal single-index models (GPLSIM). The models include the ''single-index'' models, which have beta(0) = 0. Using local linear methods, we propose estimates of the unknown parameters (alpha(0), beta(0)) and the unknown function eta(0)(.) and obtain their asymptotic distributions. Examples illustrate the models and the proposed estimation methodology.