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 c ombination of (X, Z). We generalize these models to have a nonparametr ic 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 unkno wn function: We call these generalized partially lineal single-index m odels (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 et a(0)(.) and obtain their asymptotic distributions. Examples illustrate the models and the proposed estimation methodology.