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In this work we present for the first time an analytic framework forcalculating the individual and joint distributions of the n-th most massive orn-th highest redshift galaxy cluster for a given survey characteristic allowingto formulate LCDM exclusion criteria. We show that the cumulative distributionfunctions steepen with increasing order, giving them a higher constrainingpower with respect to the extreme value statistics. Additionally, we find thatthe order statistics in mass (being dominated by clusters at lower redshifts)is sensitive to the matter density and the normalisation of the matterfluctuations, whereas the order statistics in redshift is particularlysensitive to the geometric evolution of the Universe. For a fixed cosmology,both order statistics are efficient probes of the functional shape of the massfunction at the high mass end. To allow a quick assessment of both orderstatistics, we provide fits as a function of the survey area that allowpercentile estimation with an accuracy better than two per cent. Furthermore,we discuss the joint distributions in the two-dimensional case for differentcombinations of order.  Having introduced the theory, we apply the order statistical analysis to theSPT massive cluster sample and MCXC catalogue and find that the ten mostmassive clusters in the sample are consistent with LCDM and the Tinker massfunction. In turn, by assuming the LCDM reference cosmology, order statisticscan also be utilised for consistency checks of the completeness of the observedsample and of the modelling of the survey selection function. [abridged]

Order statistics applied to the most massive and most distant galaxy clusters