## 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

Zbl 0292.60035

QRM
Full Text:

### References:

 [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 [2] P. Deheuvels, Non parametric tests of independence. in: J.P. Raoult (Ed.), Statistique Non Paramétrique Asymptotique Proceedings, Lecture Notes in Mathematics 821, Springer, Berlin, 1979, pp. 95-107. [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, (), 176-223 [5] Ferguson, T.S., Mathematical statistics: A decision theoretic approach, (1967), 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 London · Zbl 0990.62517 [9] Jogdeo, K., Concepts of dependence, (), 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. San Francisco · Zbl 0354.62038 [13] A. Lindner, A. Szimayer, A limit theorem for copulas, preprint, Technische Universität München, 2004. [14] Marshall, A.W., Copulas, marginals and joint distributions, (), 213-222 [15] McNeil, A.J.; Frey, R.; Embrechts, P., Quantitative risk management: concepts, techniques and tools, (2005), 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, (), 51-74 [18] Nelsen, R.B., An introduction to copulas, (1999), Springer New York · Zbl 0909.62052 [19] J. Nešlehová, Dependence of Non-Continuous Random Variables, Ph.D. Thesis, Carl von Ossietzky Universität Oldenburg, 2004. [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 [23] A. Tajar, M. Denuit, P. Lambert, Copula-type representations for random couples with Bernoulli margins, Discussion paper #0119, Institute of Statistics, Université Catholique de Louvain, Belgium, 2001. [24] A. Tajar, M. Mesfioui, M. Denuit, On the monotonicity of some nonparametric dependence measures with respect to concordance ordering. Discussion paper #0142, Institute of Statistics, Université Catholique de Louvain, Belgium, 2001. [25] Van der Vaart, A.W.; Wellner, J.A., Weak convergence and empirical processes: with applications to statistics, (1996), Springer New York · Zbl 0862.60002
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