Asymptotic goodness-of-fit tests for the Palm mark distribution of stationary point processes with correlated marks

TL;DR

This paper develops an asymptotic chi-squared goodness-of-fit test for the Palm mark distribution in stationary marked point processes, accommodating dependent marks and applying it to detect anisotropy in Boolean models.

Contribution

It introduces a new goodness-of-fit test for the Palm mark distribution that handles dependent marks under a beta-mixing condition, expanding applicability beyond independent mark models.

Findings

Test statistic based on empirical Palm mark distribution is mean square consistent.

The method effectively detects anisotropy in Boolean models.

Applicable to models with dependent marks and geostatistical marking.

Abstract

We consider spatially homogeneous marked point patterns in an unboundedly expanding convex sampling window. Our main objective is to identify the distribution of the typical mark by constructing an asymptotic $\chi^2$-goodness-of-fit test. The corresponding test statistic is based on a natural empirical version of the Palm mark distribution and a smoothed covariance estimator which turns out to be mean square consistent. Our approach does not require independent marks and allows dependences between the mark field and the point pattern. Instead we impose a suitable $\beta$-mixing condition on the underlying stationary marked point process which can be checked for a number of Poisson-based models and, in particular, in the case of geostatistical marking. In order to study test performance, our test approach is applied to detect anisotropy of specific Boolean models.