High-derivative parametric enhancements of nonparametric curve estimators

We suggest a method for using parametric information to modify a nonparametric estimator at the level of relatively high-order derivatives. The technique represents an alternative to methods that first fit a parametric model and then adjust it. In particular, relative to a 'nonparametric estimator with a parametric start', our estimator is not biased by the differences between parametric and nonparametric fits to low-order derivatives, since we effectively remove all the parametric information about low-order derivatives and replace it by nonparametric information. Thus, we employ parametric information only when the nonparametric information is unreliable, and do not use it elsewhere. The method has application to both nonparametric density estimation and nonparametric regression.