Generalized Partially Linear Single-Index Models

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 α T 0 X + β T 0 Z by η 0 (α T 0 X) + β T 0 Z, where η 0 (·) is an unknown function. We call these generalized partially linear single-index models (GPLSIM). The models include the "single-index" models, which have β 0 = 0. Using local linear methods, we propose estimates of the unknown parameters (α 0 , β 0 ) and the unknown function η 0 (·) and obtain their asymptotic distributions. Examples illustrate the models and the proposed estimation methodology.