High-derivative parametric enhancements of nonparametric curve estimators

We suggest a method for using parametric information to modify a nonparamet ric estimator at the level of relatively high-order derivatives. The techni que represents an alternative to methods that first fit a parametric model and then adjust it. In particular, relative to a 'nonparametric estimator w ith a parametric start', our estimator is not biased by the differences bet ween parametric and nonparametric fits to low-order derivatives, since we e ffectively remove all the parametric information about low-order derivative s and replace it by nonparametric information. Thus, we employ parametric i nformation only when the nonparametric information is unreliable, and do no t use it elsewhere. The method has application to both nonparametric densit y estimation and nonparametric regression.