A solution to a linear integral equation with an application to statistics of infinitely divisible moving averages

TL;DR

This paper presents a non-parametric estimator for the Lévy density of a stationary moving average random field, based on solving a linear integral equation, with theoretical analysis and simulation validation.

Contribution

It introduces a novel approach to estimate Lévy densities by solving a specific linear integral equation, including conditions for solution existence and error bounds.

Findings

Established conditions for solution existence and uniqueness.

Provided $L^2$-error bounds for the estimator.

Demonstrated effectiveness through simulation studies.

Abstract

For a stationary moving average random field, a non-parametric low frequency estimator of the L\'evy density of its infinitely divisible independently scattered integrator measure is given. The plug-in estimate is based on the solution $w$ of the linear integral equation $v(x) = \int_{\mathbb{R}^d} g(s) w(h(s)x)ds$, where $g,h:\mathbb{R}^d \rightarrow \mathbb{R}$ are given measurable functions and $v$ is a (weighted) $L^2$-function on $\mathbb{R}$. We investigate conditions for the existence and uniqueness of this solution and give $L^2$-error bounds for the resulting estimates. An application to pure jump moving averages and a simulation study round off the paper.