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

Abstract

The inspection of residuals is a fundamental step for investigating the quality of adjustment of a parametric model to data. For spatial point processes, the concept of residuals has been recently proposed as an empirical counterpart of the Campbell equilibrium equation for marked Gibbs point processes. The 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 residuals (raw, inverse and Pearson). On the basis of 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 that is widely used to test the null hypothesis of a homogeneous Poisson point process. © 2012 Royal Statistical Society.