Fourier methods for smooth distribution function estimation

TL;DR

This paper introduces Fourier transform techniques to analyze the asymptotic properties of kernel distribution function estimators, deriving explicit formulas for mean integrated squared error and optimal bandwidth, applicable to various kernels including superkernels.

Contribution

It provides a general framework using Fourier methods to evaluate and optimize kernel distribution function estimators, including cases with superkernels and sinc kernels.

Findings

Explicit formulas for mean integrated squared error using Fourier transforms.

Identification of conditions where kernel estimators outperform empirical distribution functions.

Analysis of optimal bandwidth sequences for different kernels.

Abstract

In this paper we show how to use Fourier transform methods to analyze the asymptotic behavior of kernel distribution function estimators. Exact expressions for the mean integrated squared error in terms of the characteristic function of the distribution and the Fourier transform of the kernel are employed to obtain the limit value of the optimal bandwidth sequence in its greatest generality. The assumptions in our results are mild enough so that they are applicable when the kernel used in the estimator is a superkernel, or even the sinc kernel, and this allows to extract some interesting consequences, as the existence of a class of distributions for which the kernel estimator achieves a first-order improvement in efficiency over the empirical distribution function.