  The paper considers probability distribution, density, conditional
distribution and density and conditional moments as well as their kernel
estimators in spaces of generalized functions. This approach does not require
restrictions on classes of distributions common in nonparametric estimation.
Density in usual function spaces is not well-posed; this paper establishes
existence and well-posedness of the generalized density function. It also
demonstrates root-n convergence of the kernel density estimator in the space of
generalized functions. It is shown that the usual kernel estimator of the
conditional distribution converges at a parametric rate as a random process in
the space of generalized functions to a limit Gaussian process regardless of
pointwise existence of the conditional distribution. Conditional moments such
as conditional mean are also be characterized via generalized functions.
Convergence of the kernel estimators to the limit Gaussian process is shown to
hold as long as the appropriate moments exist.
