Residuals and goodness-of-fit tests for stationary marked Gibbs point processes

TL;DR

This paper develops asymptotic properties of residuals for stationary marked Gibbs point processes and introduces goodness-of-fit tests with controlled Type-I error, extending classical methods like quadrat counting.

Contribution

It provides the first comprehensive analysis of residuals' asymptotic behavior for stationary marked Gibbs processes and proposes new goodness-of-fit tests based on these residuals.

Findings

Proves consistency and asymptotic normality of residuals

Develops goodness-of-fit tests with controlled Type-I error

Extends quadrat counting test to Gibbs processes

Abstract

The inspection of residuals is a fundamental step to investigate the quality of adjustment of a parametric model to data. For spatial point processes, the concept of residuals has been recently proposed by Baddeley et al. (2005) as an empirical counterpart of the {\it Campbell equilibrium} equation for marked Gibbs point processes. The present paper focuses on stationary marked Gibbs point processes and deals with asymptotic properties of residuals for such processes. In particular, the consistency and the asymptotic normality are obtained for a wide class of residuals including the classical ones (raw residuals, inverse residuals, Pearson residuals). Based on these asymptotic results, we define goodness-of-fit tests with Type-I error theoretically controlled. One of these tests constitutes an extension of the quadrat counting test widely used to test the null hypothesis of a homogeneous Poisson point process.