= Ranks as a maximal invariant under increasing transformations
On samples of distinct real observations, the labeled rank vector is a <maximal invariant> for simultaneous strictly increasing bijections of the real line. Such transformations preserve every comparison. Conversely two samples with the same ranks can be matched by a piecewise linear increasing bijection through their corresponding ordered observations, with positive-slope tails. Under a common continuous <probability distribution>, every labeled ordering is equally likely by <exchangeability>, giving distribution-free <rank tests>. This argument does not assume that all continuous distributions lie in one orbit of real-line homeomorphisms.
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