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Measures of concordance determined by \(D_4\)-invariant copulas. (English) Zbl 1075.62042

Summary: A continuous random vector \((X,Y)\) uniquely determines a copula \(C:[0,1]^2 \rightarrow [0,1]\) such that when the distribution functions of \(X\) and \(Y\) are properly composed into \(C\), the joint distribution function of \((X,Y)\) results. A copula is said to be \(D_4\)-invariant if its mass distribution is invariant with respect to the symmetries of the unit square. A \(D_4\)-invariant copula leads naturally to a family of measures of concordance having a particular form, and all copulas generating this family are \(D_4\)-invariant. The construction examined here includes Spearman’s rho and Gini’s measure of association as special cases.

MSC:

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