Global envelope tests for spatial processes

Envelope tests are a popular tool in spatial statistics, where they are used in goodness-of-fit testing. These tests graphically compare an empirical function T(r) with its simulated counterparts from the null model. However, the type I error probability α is conventionally controlled for a fixed distance r only, whereas the functions are inspected on an interval of distances I. In this study, we propose two approaches related to Barnard's Monte Carlo test for building global envelope tests on I:(1) ordering the empirical and simulated functions based on their r-wise ranks among each other, and (2) the construction of envelopes for a deviation test. These new tests allow the a priori selection of the global α and they yield p-values. We illustrate these tests using simulated and real point pattern data.