Partially Linear Additive Models with Unknown Link Functions

In this paper, we consider partially linear additive models with an unknown link function, which include single-index models and additive models as special cases. We use polynomial spline method for estimating the unknown link function as well as the component functions in the additive part. We establish that convergence rates for all nonparametric functions are the same as in one-dimensional nonparametric regression. For a faster rate of the parametric part, we need to define appropriate ‘projection’ that is more complicated than that defined previously for partially linear additive models. Compared to previous approaches, a distinct advantage of our estimation approach in implementation is that estimation directly reduces estimation in the single-index model and can thus deal with much larger dimensional problems than previous approaches for additive models with unknown link functions. Simulations and a real dataset are used to illustrate the proposed model.