ABSTRACT
Postulating strict relationships between dependent and independent variables, parametric models are, in general, parsimonious. Parameters in these models often have meaningful interpretations. On the other hand, based on minimal assumptions about the relationship, nonparametric models are flexible. However, nonparametric models lose the advantage of having interpretable parameters and may suffer from the curse of dimensionality. Often, in practice, there is enough knowledge to model some components in the regression function parametrically. For other vague and/or nuisance components, one may want to leave them unspecified. Combining both parametric and nonparametric components, a semiparametric regression model can overcome limitations in parametric and nonparametric models while maintaining advantages of having interpretable parameters and flexibility. Many specific semiparametric models have been proposed in the lit-
erature. The partial spline model (2.43) in Section 2.10 is perhaps the simplest semiparametric regression model. Other semiparametric models include the projection pursuit, single index, varying coefficients, functional linear, and shape invariant models. A projection pursuit regression model (Friedman and Stuetzle 1981) assumes that
yi = β0 +
fk(β T k xi) + ǫi, i = 1, . . . , n, (8.1)
where x are independent variables, β0 and βk are parameters, and fk are nonparametric functions. A partially linear single index model (Carroll, Fan, Gijbels and Wand 1997, Yu and Ruppert 2002) assumes that
yi = β T 1 si + f(β
T 2 ti) + ǫi, i = 1, . . . , n, (8.2)
where s and t are independent variables, β1 and β2 are parameters, and
Tibshirani 1993) assumes that
yi = β1 +
uikfk(xik) + ǫi, i = 1, . . . , n, (8.3)
where xk and uk are independent variables, β1 is a parameter, and fk are nonparametric functions. The semiparametric linear and nonlinear regression models in this
chapter include all the foregoing models as special cases. The general form of these models provides a framework for unified estimation, inference, and software development.
