Deconvolution of distribution functions without integral transforms

TL;DR

The paper addresses the problem of recovering the distribution function $F_X$ from blurred observations $Y=X+\varepsilon$ under independent additive errors, focusing on distribution-function-based deconvolution rather than densities. Its core idea is to convert the first-kind convolution equation into a second-kind integral equation and solve it via Picard iterations, yielding Neumann-sum representations that enable unbiased plug-in estimation from a $Y$-sample. The authors obtain explicit inverse representations in right-lateral discrete noise settings, extend the framework to non-discrete $X$ through operator-theoretic constructions, and explore convergence and invertibility properties of the resulting deconvolution operators. The work offers a principled, transform-free pathway to deconvolution in the distribution-function domain and outlines future directions for Fourier-domain analysis and broader applicability.

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.

Figures (6)

• Figure 1: Plots for target d.f. $F_X$, blurred d.f. $F_Y$ and plug-in estimator of inverse, in various purely discrete setups that match Corollary \ref{['SatzdiskrDekWkeitsfkt']}. In all these cases, $s=1$, $\xi_0=z_0=0$ and $\Theta \lbrace \ddot p_Y \rbrace = (F_\varepsilon \lbrace 0 \rbrace )^{-1} F_Y$, so that the estimator is given by $\mathfrak{F}_n := (n F_\varepsilon \lbrace 0 \rbrace )^{-1} \sum_{i=1}^n \Theta \lbrace \gamma \lbrace \ddot u_{\varepsilon, +} \rbrace \rbrace (\cdot - Y_i)$, for an i.i.d. sample $Y_1, \hdots, Y_n$ of size $n \in \mathbb{N}$.
• Figure 2: The above plots consider situations that match Corollary \ref{['BspXdiskrZstet']}, for distributions with $s=1$, $\xi_0 = z_0 = 0$ and $\zeta_\ell := \xi_{\ell+1}$. Then, defining $\ddot P_{Y, n}(\ell) := ( F_\varepsilon(1) )^{-1} F_Y(1+\ell, n)$, an unbiased deconvolution estimator for $F_X$ is given by $\mathfrak{F}_n := \Theta \lbrace \ddot P_{Y, n} \rbrace \ast \Theta \lbrace \gamma \lbrace \ddot U_{\varepsilon, +} \rbrace \rbrace$. To decrease computational effort, it is helpful to observe that $\mathfrak{F}_n(\xi) = (nF_\varepsilon(1))^{-1} \sum_{i=1}^n \sum_{j=0}^{ \left\lfloor \xi \right\rfloor } \Theta \lbrace \gamma \lbrace \ddot U_{\varepsilon, +} \rbrace \rbrace (1+j-Y_i)$, for $\xi \in \mathbb{R}$ and an i.i.d. sample $Y_1, \hdots, Y_n$.
• Figure 3: Plots for the deconvolution of a right-lateral continuous target d.f. $F_X$ from a blurred d.f. $F_Y$. The plug-in estimator $\mathfrak{F}_n$ for $F_X$ was constructed from Corollary \ref{['BspXbelZdiskr']}, based on a $Y$-sample of size $n \in \mathbb{N}$. In all cases, $(z_0, t) = (0, 1)$, so that $P_Y = F_Y$.
• Figure 4: Deconvolution of a bilateral d.f. $F_X$ from a blurred d.f. $F_Y$, through a plug-in estimator $\mathfrak{F}_n$ that was constructed from a $Y$-sample of size $n \in \mathbb{N}$, with the aid of Corollary \ref{['BspXbelZdiskr']}. Again, $(z_0, t) = (0, 1)$ and hence $P_Y = F_Y$, in all scenarios.
• Figure 5: Plots for the deconvolution of a standard normal target from normally distributed errors. Notice how an increasement in the error variance substantially decreases the rate of convergence of $\mathfrak{F}(\cdot, m)$ to $F_X$. Furthermore, in case of non-centered errors, on some segments of the real axis apparently no convergence can be expected.

Theorems & Definitions (45)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.1: deconvolution I
• proof : Proof of Theorem \ref{['Theo2025073001']}
• Corollary 3.1.1: deconvolution II
• proof
• Corollary 3.1.2: left-bounded monotonic $\mathbb{T}_X$ and left-bounded countable $\mathbb{T}_\varepsilon$
• proof : Proof of Corollary \ref{['SatzdiskrDekWkeitsfkt']}