Statistics for Poisson models of overlapping spheres

TL;DR

This paper introduces nonparametric estimators for the radius distribution in a Poisson Boolean model with spherical grains, demonstrating their consistency, asymptotic normality, and providing explicit variance expressions.

Contribution

It proposes a new family of ratio-unbiased, asymptotically consistent estimators based on observed distances and radii for the Poisson Boolean model.

Findings

Estimators are ratio-unbiased and consistent.

Asymptotic normality is established under certain conditions.

Explicit integral expression for asymptotic variance.

Abstract

The paper considers the stationary Poisson Boolean model with spherical grains and proposes a family of nonparametric estimators for the radius distribution. These estimators are based on observed distances and radii, weighted in an appropriate way. They are ratio-unbiased and asymptotically consistent for growing observation window. It is shown that the asymptotic variance exists and is given by a fairly explicit integral expression. Asymptotic normality is established under a suitable integrability assumption on the weight function. The paper also provides a short discussion of related estimators as well as a simulation study.