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 upon 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. These are the {\it generalized partially linear single-index models}. The models also generalize the ``single-index'' models, which have $\beta_03D0$. Using local linear methods, estimates of the unknown parameters $(\alpha_0,\beta_0)$ and the unknown function $\eta_0(.)$ are proposed, and their asymptotic distributions obtained. An example illustrates the algorithms and the models.