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.


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


Zbl 0292.60035


Full Text: DOI


[1] Capéraà, P.; Genest, C., Spearman’s \(\rho\) is larger than Kendall’s \(\tau\) for positively dependent random variables, J. Nonparametric Statist., 2, 183-194 (1993) · Zbl 1360.62294
[3] Denuit, M.; Lambert, P., Constraints on concordance measures in bivariate discrete data, J. Multivariate Anal., 93, 40-57 (2005) · Zbl 1095.62065
[4] Embrechts, P.; McNeil, A. J.; Straumann, D., Correlation and dependence in risk management: properties and pitfalls, (Dempster, M. A.H., Risk ManagementValue at Risk and Beyond (2002), Cambridge University Press: Cambridge University Press Cambridge), 176-223
[5] Ferguson, T. S., Mathematical Statistics: A Decision Theoretic Approach (1967), Academic Press: Academic Press New York · Zbl 0153.47602
[6] Hoeffding, W., Maßstabinvariante Korrelationstheorie, Schrift. Math. Seminars Inst. Angew. Math. Univ. Berlin, 5, 3, 181-233 (1940) · JFM 66.0649.02
[7] Hoeffding, W., Maßstabinvariante Korrelationstheorie für diskontinuierliche Verteilungen, Arch. Math. Wirtschafts- und Sozialforschung, VII, 2, 4-70 (1940)
[8] Joe, H., Multivariate Models and Dependence Concepts (1997), Chapman & Hall: Chapman & Hall London · Zbl 0990.62517
[9] Jogdeo, K., Concepts of dependence, (Kotz, S.; Johnson, N. L., Encyclopedia of Statistical Sciences (1982), Wiley: Wiley New York), 324-334
[10] Kruskal, W. H., Ordinal measures of association, J. Amer. Statist. Assoc., 53, 814-861 (1958) · Zbl 0087.15403
[11] Lehmann, E. L., Some concepts of dependence, Ann. of Math. Statist., 37, III, 1137-1153 (1966) · Zbl 0146.40601
[12] Lehmann, E. L., Nonparametrics: Statistical Methods Based on Ranks (1975), Holden-Day, Inc.: Holden-Day, Inc. San Francisco · Zbl 0354.62038
[14] Marshall, A. W., Copulas, marginals and joint distributions, (Rüschendorf, L.; Schweizer, B.; Taylor, M. D., Distributions with Fixed Marginals and Related Topics (1996), Institute of Mathematical Statistics: Institute of Mathematical Statistics Hayward, CA), 213-222
[15] McNeil, A. J.; Frey, R.; Embrechts, P., Quantitative Risk Management: Concepts, Techniques and Tools (2005), Princeton University Press: Princeton University Press Princeton · Zbl 1089.91037
[16] Mesfioui, M.; Tajar, A., On the properties of some nonparametric concordance measures in the discrete case, J. Nonparametric Statist., 17, 541-554 (2005) · Zbl 1135.60303
[17] Nelsen, R. B., Copulas and association, (Dall’Aglio, G.; Kotz, S.; Salinetti, G., Advances in Probability Distributions with Given Marginals (1991), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 51-74
[18] Nelsen, R. B., An Introduction to Copulas (1999), Springer: Springer New York · Zbl 0909.62052
[20] Scarsini, M., On measures of concordance, Stochastica, 8, 201-218 (1984) · Zbl 0582.62047
[21] Schweizer, B.; Sklar, A., Operation on distribution functions not derivable from operations on random variables, Studia Math., 52, 43-52 (1974) · Zbl 0292.60035
[22] Schweizer, B.; Wolff, E. F., On nonparametric measures of dependence for random variables, Ann. Statist., 9, 870-885 (1981) · Zbl 0468.62012
[25] Van der Vaart, A. W.; Wellner, J. A., Weak Convergence and Empirical Processes: With Applications to Statistics (1996), Springer: Springer New York · Zbl 0862.60002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.