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Measures of concordance determined by \(D_4\)-invariant measures on \((0,1)^2\). (English) Zbl 1081.62032

Summary: A measure, \(\mu\), on \((0,1)^2\) is said to be \(D_4\)-invariant if its value for any Borel set is invariant with respect to the symmetries of the unit square. A function, \(\kappa\), generated in a certain way by a measure, \(\mu\), on \((0,1)^2\) is shown to be a measure of concordance if and only if the generating measure is positive, regular, \(D_4\)-invariant, and satisfies certain inequalities. The construction examined here includes Blomqvist’s beta as a special case.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H20 Measures of association (correlation, canonical correlation, etc.)
60A10 Probabilistic measure theory
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