Deconvolution of distribution functions without integral transforms

TL;DR

This paper introduces a novel approach to deconvolving distribution functions affected by additive noise without using integral transforms, enabling unbiased estimation from sample data.

Contribution

It develops a new method transforming a first kind to a second kind integral equation for deconvolution directly on distribution functions, avoiding traditional integral transform techniques.

Findings

Provides an explicit representation for $F_X$ in terms of observable quantities.

Extends the method to cases where $X$ is not discrete.

Offers an approximation for $F_X$ as a Neumann sum with theoretical and visual analysis.

Abstract

We study the recovery of the distribution function $F_X$ of a random variable $X$ that is subject to an independent additive random error $\varepsilon$. To be precise, it is assumed that the target variable $X$ is available only in the form of a blurred surrogate $Y = X + \varepsilon$. The distribution function $F_Y$ then corresponds to the convolution of $F_X$ and $F_\varepsilon$, so that the reconstruction of $F_X$ is some kind of deconvolution problem. Those have a long history in mathematics and various approaches have been proposed in the past. Most of them use integral transforms or matrix algorithms. The present article avoids these tools and is entirely confined to the domain of distribution functions. Our main idea relies on a transformation of a first kind to a second kind integral equation. Thereof, starting with a right-lateral discrete target and error variable, a representation for $F_X$ in terms of available quantities is obtained, which facilitates the unbiased estimation through a $Y$-sample. It turns out that these results even extend to cases in which $X$ is not discrete. Finally, in a general setup, our approach gives rise to an approximation for $F_X$ as a certain Neumann sum. The properties of this sum are briefly examined theoretically and visually. The paper is concluded with a short discussion of operator theoretical aspects and an outlook on further research. Various plots underline our results and illustrate the capabilities of our functions with regard to estimation.