Takacs Fiksel method for stationary marked Gibbs point processes

TL;DR

This paper introduces the Takacs-Fiksel method for estimating parameters of stationary marked Gibbs point processes, leveraging the Georgii-Nguyen-Zessin formula, with theoretical guarantees and simulation validation.

Contribution

It develops a flexible estimation procedure based on test functions, providing conditions for consistency and asymptotic normality in Gibbs models.

Findings

The method is applicable to exponential family models.

Asymptotic properties are established under certain conditions.

Simulation results support the theoretical findings.

Abstract

This paper studies a method to estimate the parameters governing the distribution of a stationary marked Gibbs point process. This procedure, known as the Takacs-Fiksel method, is based on the estimation of the left and right hand sides of the Georgii-Nguyen-Zessin formula and leads to a family of estimators due to the possible choices of test functions. We propose several examples illustrating the interest and flexibility of this procedure. We also provide sufficient conditions based on the model and the test functions to derive asymptotic properties (consistency and asymptotic normality) of the resulting estimator. The different assumptions are discussed for exponential family models and for a large class of test functions. A short simulation study is proposed to assess the correctness of the methodology and the asymptotic results.