Nonparametric functionals as generalized functions

TL;DR

This paper introduces a generalized function framework for nonparametric estimation of distributions and moments, ensuring well-posedness and convergence properties without restrictive assumptions.

Contribution

It establishes the existence, well-posedness, and convergence of kernel estimators for distributions and moments within the space of generalized functions, broadening nonparametric analysis.

Findings

Kernel density estimator converges at root-n rate in generalized functions space.

Conditional distribution estimators converge to a Gaussian process regardless of pointwise existence.

Generalized functions enable well-posedness of density and moments in nonparametric estimation.

Abstract

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.