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We propose two model-free, permutation-based tests of independence between apair of random variables. The tests can be applied to samples from anybivariate distribution: continuous, discrete or mixture of those, with lighttails or heavy tails, \ldots The tests take advantage of the recent developmentof the global envelope tests in the context of spatial statistics. Apart fromthe broad applicability of the tests, their main benefit lies in the graphicalinterpretation of the test outcome: in case of rejection of the null hypothesisof independence, the combinations of quantiles in the two marginals areindicated for which the deviation from independence is significant. Thisinformation can be used to gain more insight into the properties of theobserved data and as a guidance for proposing more complicated models andhypotheses. We assess the performance of the proposed tests in a simulationstudy and compare them to several well-established tests of independence.Furthermore, we illustrate the use of the tests and the interpretation of thetest outcome in two real datasets consisting of meteorological reports (dailymean temperature and total daily precipitation, having an atomic component at 0millimeters) and road accidents reports (type of road and the weatherconditions, both variables having categorical distribution).

Graphical tests of independence for general distributions