Goodness-of-fit test of the mark distribution in a point process with non-stationary marks

Three procedures for testing the hypothesis that the marks of the point process belong to a given family of distributions, where the marks are independent conditionally on an unknown non-stationary parametric field θ(x), are described. The unknown parametric field was estimated by a kernel estimator and the estimate included in the procedures. The asymptotic distribution of the kernel estimator was derived because it is required in some of the procedures to take into account the estimation uncertainty. The procedures were compared by a simulation study and applied to two real datasets. First, the terminal dates of Maya sites and second, the heights of trees that have fallen during storms.