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Generalized partially linear single-index models. (English) Zbl 0890.62053

Summary: 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 \(\alpha^T_0 X+ \beta^T_0Z\) by \(\eta_0 (\alpha^T_0X) +\beta^T_0Z\), where \(\eta_0 (\cdot)\) is an unknown function. We call these generalized partially linear 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 \(\eta_0 (\cdot)\) and obtain their asymptotic distributions. Examples illustrate the models and the proposed estimation methodology.

MSC:

62J12 Generalized linear models (logistic models)
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62E20 Asymptotic distribution theory in statistics
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