Non-parametric asymptotic statistics for the Palm mark distribution of \beta-mixing marked point processes

TL;DR

This paper develops a non-parametric asymptotic method to identify the distribution of marks in spatial point processes, using a -goodness-of-fit test that accommodates dependence and -mixing conditions.

Contribution

It introduces a novel -goodness-of-fit test for the Palm mark distribution in -mixing marked point processes, extending existing methods to dependent marks and spatial correlations.

Findings

Test performs well in simulations with varying spatial correlations.

Method is applicable to Poisson-based and geostatistical marking models.

Consistent covariance estimation under -mixing conditions.

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. Our method needs a central limit theorem for \beta-mixing random fields which is proved by extending Bernstein's blocking technique to non-cubic index sets and seems to be of interest in its own right. By large-scale model-based simulations the performance of our test is studied in dependence of the model parameters which determine the range of spatial correlations.