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

摘要

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