For a pragmatic scientist the conclusion of Fisher (1922), to “confine ourselves to those forms that we know how to handle,” must have an irresistible attractive power. Indeed, we know that the nonparametric smoothing task is hard, especially in high dimensions. So why not come back to parametrics, at least partially? A parametric together with a nonparametric component may handle the model building even better than just the nonparametric or the parametric approach! In this chapter I present approaches from both views. The discussed models incorporate both parametric and nonparametric components and are therefore called semiparametric models.

Three topics are addressed. First, the estimation of parameters in a partial linear model. Second, the comparison of individual curves in a shape-invariant context. Third, a method is proposed to check the appropriateness of parametric regression curves by comparison with a nonparametric smoothing estimator.