Abstract :
Nonparametric kernel estimation of density and conditional mean is widely used,
but many of the pointwise and global asymptotic results for the estimators are not
available unless the density is continuous and appropriately smooth; in kernel estimation
for discrete-continuous cases smoothness is required for the continuous
variables+ Nonsmooth density and mass points in distributions arise in various
situations that are examined in empirical studies; some examples and explanations
are discussed in the paper+ Generally, any distribution function consists of
absolutely continuous, discrete, and singular components, but only a few special
cases of nonparametric estimation involving singularity have been examined in
the literature, and asymptotic theory under the general setup has not been developed+
In this paper the asymptotic process for the kernel estimator is examined
by means of the generalized functions and generalized random processes approach;
it provides a unified theory because density and its derivatives can be defined as
generalized functions for any distribution, including cases with singular components+
The limit process for the kernel estimator of density is fully characterized
in terms of a generalized Gaussian process+ Asymptotic results for the Nadaraya–
Watson conditional mean estimator are also provided+
