Deconvolution of Arbitrary Distribution Functions and Densities
Henrik Kaiser
TL;DR
This work develops a general framework for deconvolution under additive measurement error by symmetrizing the error distribution to obtain a cf in $[0,1]$ and expressing the target cf $\Phi_X$ as a convergent geometric series. The authors introduce the deconvolution function $\mathfrak D(\xi,m)$ (and density $\mathfrak d(\xi,m)$) as finite truncations of this series, with Fourier-type representations enabling analysis of convergence to the true distribution $F_X$ and density $f_X$. They establish fundamental properties, moment relations, and integral representations, and provide conditions under which $\mathfrak D(\cdot,m)$ and $\mathfrak d(\cdot,m)$ converge (pointwise, weakly, or in density) to $F_X$ and $f_X$, including scenarios with zeros of $\Phi_\varepsilon$. The results yield a principled, plug-in pathway for nonparametric deconvolution estimators from $Y$-samples, and point to rich avenues for bias-variance analysis and rate results using Fourier-type and complex-analytic techniques.
Abstract
In this article we recover the distribution function (and possible density) of an arbitrary random variable that is subject to an additive measurement error. This problem is also known as deconvolution and has a long tradition in mathematics. We show that the model under consideration always can be transformed to a model with a symmetric error variable, whose characteristic function has its values in the unit interval. As a consequence, the characteristic function of the target variable turns out as the limit of a geometric series. By truncation of this series, an approximation for the associated distribution function (and density) is established. The convergence properties of these approximations are examined in detail across diverse setups.