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Given a set of incomplete observations, we study the nonparametric problem oftesting whether data are Missing Completely At Random (MCAR). Our firstcontribution is to characterise precisely the set of alternatives that can bedistinguished from the MCAR null hypothesis. This reveals interesting and novellinks to the theory of Fr\'echet classes (in particular, compatibledistributions) and linear programming, that allow us to propose MCAR tests thatare consistent against all detectable alternatives. We define anincompatibility index as a natural measure of ease of detectability, establishits key properties, and show how it can be computed exactly in some cases andbounded in others. Moreover, we prove that our tests can attain the minimaxseparation rate according to this measure, up to logarithmic factors. Ourmethodology does not require any complete cases to be effective, and isavailable in the R package MCARtest.

Optimal nonparametric testing of Missing Completely At Random, and its connections to compatibility