Abstract
Invariance properties play an important role in statistics. But there exist a lot of invariance concepts partially motivated by a criterion of meaningfulness of quantitative relations (see Suppes [1959] or Pfanzagl [1971]). If the values of a function are equal for two distinct arguments and this equality is invariant with respect to a group of transformations we call the function comparison-invariant. If the value of a function does not change the function will be called absolute-invariant. We want to show by group-theoretic means that statistical functions comparison-invariant with respect to the group of the increasing, continuous transformations on R necessarily possess the stronger property of absolute-invariance. This result has several consequences for the possibility of constructing statistical functions for ordinal data.
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