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Nonparametric inference for Poisson-Laguerre tessellations

Thomas van der Jagt, Geurt Jongbloed, Martina Vittorietti

TL;DR

This work develops two nonparametric estimators for the distribution function $F$ governing Poisson-Laguerre tessellations, relying on observed extreme points and Laguerre cells. The first estimator uses a dependent thinning of the point process, yielding a strongly consistent inverse estimator $\hat{F}_n^0$, while the second leverages the volume-biased weight distribution to yield another inverse estimator $\hat{F}_n$, with consistency results for both approaches. A stereological extension connects planar observations to higher-dimensional weights via an Abel-type inversion, enabling estimation of the higher-dimensional distribution $H$ through monotone/isotonic regression. The paper also includes simulations validating the estimators in both direct and sectional settings and discusses practical considerations such as edge effects and numerical stability. Overall, the results provide provable, nonparametric inference tools for us to recover the underlying weight distribution from Laguerre tessellations and their sections.

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.

Nonparametric inference for Poisson-Laguerre tessellations

TL;DR

This work develops two nonparametric estimators for the distribution function governing Poisson-Laguerre tessellations, relying on observed extreme points and Laguerre cells. The first estimator uses a dependent thinning of the point process, yielding a strongly consistent inverse estimator , while the second leverages the volume-biased weight distribution to yield another inverse estimator , with consistency results for both approaches. A stereological extension connects planar observations to higher-dimensional weights via an Abel-type inversion, enabling estimation of the higher-dimensional distribution through monotone/isotonic regression. The paper also includes simulations validating the estimators in both direct and sectional settings and discusses practical considerations such as edge effects and numerical stability. Overall, the results provide provable, nonparametric inference tools for us to recover the underlying weight distribution from Laguerre tessellations and their sections.

Abstract

In this paper, we consider statistical inference for Poisson-Laguerre tessellations in . The object of interest is a distribution function which uniquely determines the intensity measure of the underlying Poisson process. Two nonparametric estimators for 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.
Paper Structure (13 sections, 19 theorems, 105 equations, 6 figures, 1 table)

This paper contains 13 sections, 19 theorems, 105 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Poisson-Laguerre tessellations
  4. Inference via a dependent thinning
  5. Definition of an estimator
  6. Consistency
  7. Inference via the volume-biased weight distribution
  8. Definition of an estimator
  9. Consistency
  10. Stereology
  11. Simulations
  12. Proofs
  13. Computational formula

Key Result

Lemma 1

Let $B \in \mathcal{B}(\mathbb{R}^d)$, $y\in \mathbb{R}^d$ and $z \geq 0$, the intensity measure $\Lambda^y$ of $\eta^y$ satisfies:

Figures (6)

  • Figure 1: Left: A realization of a planar Poisson-Laguerre tessellation. Cells are colored according to their area. Right: The corresponding realization of extreme points. Around each point there is a circle with radius proportional to the weight of the point.
  • Figure 2: Visualization of the crystallization process. From left to right, the crystallization process is shown at times $t = 60$, $t=80$, $t=120$ and $t=280$.
  • Figure 3: Left: A realization of $\hat{G}_n$. Right: The corresponding realization of $\hat{F}_n^0$. The actual underlying $F$ is equal to the CDF of a uniform distribution on $(0, 1)$.
  • Figure 4: A comparison of the plugin estimator $H_n$ and the isotonic estimator $\hat{H}_n$. The actual underlying $H$ is equal to the CDF of a uniform distribution on $(0, 1)$.
  • Figure 5: Simulation results for $\hat{F}_n^0$ and $\hat{F}_n$, where $F$ is given by (\ref{['true_F_option1']}), with $M=1$ (Left) and $M=3$ (Right).
  • ...and 1 more figures

Theorems & Definitions (41)

  • Definition 1
  • Definition 2
  • Example 1
  • Lemma 1
  • Theorem 3: Mecke equation
  • proof : Proof of Lemma \ref{['dependent_thinning_measure']}
  • Theorem 4
  • Theorem 5: Theorem 1.3.3. in Pachpatte1998
  • proof : Proof of Theorem \ref{['thm_G_identifiable']}
  • Definition 6: First inverse estimator of $F$
  • ...and 31 more