A Note on the Maximum Difference Between Schweizer and Wolff's $\sigma$ and the Absolute Value of Spearman's $\rho$
Subject Areas : Transactions on Fuzzy Sets and Systems
1 - Department of Mathematics, University of Almer´ıa, Almer´ıa, Spain.
Keywords: Copula, Schweizer and Wolff's $sigma$, Spearman's $rho$,
Abstract :
In this note we correct an error on the possible maximum difference between the (measure of dependence) Schweizer and Wolff’s σ and the absolute value of the (measure of concordance) Spearman’s ρ given in [8]. Moreover, we provide a possible value for that possible, leaving its proof as an open problem.
[1] G. Beliakov, A. Pradera and T. Calvo, Aggregation Functions: A Guide for Practitioners, Studies in Fuzziness and Soft, Springer, Berlin, (2007).
[2] F. Durante and C. Sempi, Principles of Copula Theory, Chapman & Hall/CRC, Boca Raton, (2016).
[3] M. Grabisch, J. L. Marichal, R. Mesiar and E. Pap, Aggregation Functions, Encyclopedia of Mathematics and its Applications, 127, Cambridge University Press, Cambridge, (2009).
[4] P. R. Halmos, Measure Theory, Springer, New York, (1974).
[5] R. B. Nelsen, An Introduction to Copulas, Second Edition. Springer, New York, (2006).
[6] A. Renyi, On measures of dependence, Acta Math. Acad. Sci. Hungar., 10 (1959), 441-451.
[7] M. Scarsini, On measures of concordance, Stochastica, 8 (1984), 201-218.
[8] B. Schweizer and E. F. Wol , On nonparametric measures of dependence for random variables, Ann. Statist., 9 (1981), 879-885.
[9] A. Sklar, Fonctions de repartition a n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, 8 (1959), 229-231.
[10] C. Spearman, The proof and measurement of association between two things, Am. J. Psychol., 15 (1904), 72-101.
[11] E. F. Wol , Measures of dependence derived from copulas, Ph.D. Thesis, University of Massachussetts, Amherst, USA, (1977).
