Nonparametric inference for Poisson-Laguerre tessellations

TL;DR

This paper introduces and proves the consistency of two nonparametric estimators for the distribution function of Poisson-Laguerre tessellations, applicable in both full space and sectional observation settings.

Contribution

It proposes novel nonparametric estimators for the distribution function of Poisson-Laguerre tessellations and establishes their strong consistency as observation windows expand.

Findings

Two nonparametric estimators are strongly consistent.

Estimators depend only on generating points and corresponding cells.

Results extend to sectional Poisson-Laguerre tessellations.

Abstract

In this paper, we consider statistical inference for Poisson-Laguerre tessellations in $\mathbb{R}^d$. The object of interest is a distribution function $F$ which uniquely determines the intensity measure of the underlying Poisson process. Two nonparametric estimators for $F$ are introduced which depend only on the points of the Poisson process which generate non-empty cells and the actual cells corresponding to these points. The proposed estimators are proven to be strongly consistent, as the observation window expands unboundedly to the whole space. We also consider a stereological setting, where one is interested in estimating the distribution function associated with the Poisson process of a higher dimensional Poisson-Laguerre tessellation, given that a corresponding sectional Poisson-Laguerre tessellation is observed.