Global envelope tests for spatial processes

Mari Myllymäki*, Tomáš Mrkvička, Pavel Grabarnik, Henri Seijo, Ute Hahn

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

122 Citations (Scopus)


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: ordering the empirical and simulated functions on the basis of their r-wise ranks among each other, and the construction of envelopes for a deviation test. These new tests allow the a priori choice of the global α and they yield p-values. We illustrate these tests by using simulated and real point pattern data.

Original languageEnglish
Pages (from-to)381-404
Number of pages24
JournalJournal of the Royal Statistical Society. Series B: Statistical Methodology
Issue number2
Publication statusPublished - 1 Mar 2017
MoE publication typeA1 Journal article-refereed


  • Deviation test
  • Functional depth
  • Global envelope test
  • Goodness-of-fit test
  • Monte Carlo p-value
  • Spatial point pattern


Dive into the research topics of 'Global envelope tests for spatial processes'. Together they form a unique fingerprint.

Cite this