# An Inverse Problem for Infinitely Divisible Moving Average Random Fields

## TL;DR

This paper addresses the nonparametric estimation of Lévy characteristics in infinitely divisible moving average random fields using three different methods, providing theoretical error bounds and simulation comparisons.

## Contribution

It introduces three novel estimation methods for Lévy densities in random fields and analyzes their theoretical performance and practical effectiveness.

## Key findings

- All three methods provide consistent $L^2$-error bounds.
- Numerical simulations compare the performance of the methods.
- The Fourier-based approach shows promising accuracy in simulations.

## Abstract

Given a low frequency sample of an infinitely divisible moving average random field $\{\int_{\mathbb{R}^d} f(x-t)\Lambda(dx); \ t \in \mathbb{R}^d \}$ with a known simple function $f$, we study the problem of nonparametric estimation of the L\'{e}vy characteristics of the independently scattered random measure $\Lambda$. We provide three methods, a simple plug-in approach, a method based on Fourier transforms and an approach involving decompositions with respect to $L^2$-orthonormal bases, which allow to estimate the L\'{e}vy density of $\Lambda$. For these methods, the bounds for the $L^2$-error are given. Their numerical performance is compared in a simulation study.