Multiple Monte Carlo testing,with applications in spatial point processes

The rank envelope test (Myllymäki et al. in J R Stat Soc B, doi: 10.1111/rssb.12172, 2016) is proposed as a solution to the multiple testing problem for Monte Carlo tests. Three different situations are recognized: (1) a few univariate Monte Carlo tests, (2) a Monte Carlo test with a function as the test statistic, (3) several Monte Carlo tests with functions as test statistics. The rank test has correct (global) type I error in each case and it is accompanied with a p-value and with a graphical interpretation which determines subtests and distances of the used test function(s) which lead to the rejection at the prescribed significance level of the test. Examples of null hypotheses from point process and random set statistics are used to demonstrate the strength of the rank envelope test. The examples include goodness-of-fit test with several test functions, goodness-of-fit test for a group of point patterns, test of dependence of components in a multi-type point pattern, and test of the Boolean assumption for random closed sets. A power comparison to the classical multiple testing procedures is given.