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On rank correlation measures for non-continuous random variables. (English) Zbl 1107.62047

Summary: For continuous random variables, many dependence concepts and measures of association can be expressed in terms of the corresponding copula only and are thus independent of the marginal distributions. This interrelationship generally fails as soon as there are discontinuities in the marginal distribution functions. We consider an alternative transformation of an arbitrary random variable to a uniformly distributed one. Using this technique, the class of all possible copulas in the general case is investigated. In particular, we show that one of its members, the standard extension copula introduced by B. Schweizer and A. Sklar [Stud. Mat. 52, 43–52 (1974; Zbl 0292.60035)], captures the dependence structures in an analogous way the unique copula does in the continuous case. Furthermore, we consider measures of concordance between arbitrary random variables and obtain generalizations of Kendall’s tau and Spearman’s rho that correspond to the sample version of these quantities for empirical distributions.

MSC:

62H20 Measures of association (correlation, canonical correlation, etc.)
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62G30 Order statistics; empirical distribution functions

Citations:

Zbl 0292.60035

Software:

QRM
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References:

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