Invariant functions on measurement structures (Q1896623)

Since the publication of \textit{S. Stevens}' celebrated articles [Science 103, 677-680 (1946); S. Stevens (ed.), Handb. Exp. Psychol., 1-49 (1951)]\ the relationship between measurement structures and appropriate statistics has been an outstanding problem. In an attempt to explicate Stevens' idea of ``permissible statistics'', \textit{E. W. Adams} et al. [Psychometrika 30, 99-127 (1967)]\ have studied absolute invariance, reference invariance, and comparison invariance of real-valued functions called ``generalized operations on measurements'' with respect to measurement structures (called ``numerical assignment systems''). \textit{J. Pfanzagl} [Theory of measurement. (1968; Zbl 0186.536)], while developing a theory of ``meaningful relations'', uses the concept of comparison invariance to define ``meaningful statistics'', which are real-valued functions on measurement structures. And \textit{I. Klein} [Stat. Hefte 26, 313-320 (1985; Zbl 0592.62003)]\ has studied absolute invariance, reference invariance, and comparison invariance of real- valued functions on statistical distributions. We extend the concepts of absolute invariance, reference invariance, and comparison invariance to vector-valued functions on relational systems which are range systems of measurement structures. We also study the characterization and interrelationships of these concepts of invariance. The results can serve as a basis for a more general treatment of the problem of appropriate (or ``meaningful'') statistics which include multidimensional statistics.