Mutual association measures

  • Claudio G. BorroniEmail author
Original Paper


Given two continuous variables X and Y with copula C,  many attempts were made in the literature to provide a suitable representation of the set of all concordance measures of the couple (XY), as defined by some axioms. Some papers concentrated on the need to let such measures vanish not only at independence but, more generally, at indifference, i.e. in lack of any dominance of concordance or discordance in (XY). The concept of indifference (or reflection-invariance in the language of copulas) led eventually to full characterizations of some well-defined subsets of measures. However, the built classes failed to contain some known measures, which cannot be regarded as averaged “distances” of C from given reference copulas (markedly, the Frechét–Hoeffding bounds). This paper, then, proposes a method to enlarge the representation of concordance measures, so as to include such elements, denoted here as mutual, which arise from averaged multiple comparisons of the copula C itself. Mutual measures, like Kendall’s tau and a recent proposal based on mutual variability, depend on the choice of a generator function, which needs some assumptions listed here. After defining some natural estimators for the elements of the enlarged class, the assumptions made on the generator are shown to guarantee that those statistics possess desirable sample properties, such as asymptotic normality under independence. Distributions under contiguous alternatives and relative efficiencies, in addition, are derived under mild assumptions on the copula. The paper provides several examples, giving further insight to some comparisons formerly conducted in the literature only via simulation.


Copula-based measures of association Measures of concordance Rank correlation measures Contiguous alternatives Asymptotic relative efficiency 



Special thanks go to two anonymous referees for their suggestions to improve a first version of this paper.


  1. Behboodian J, Dolati A, Úbeda-Flores M (2005) Measures of association based on average quadrant dependence. J Probab Stat Sci 3:161–173Google Scholar
  2. Billingsley P (1999) Convergence of probability measures, 2nd edn. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  3. Borroni CG (2013) A new rank correlation measure. Stat Pap 54:255–270MathSciNetCrossRefzbMATHGoogle Scholar
  4. Borroni CG, Cifarelli DM (2016) Some maximum-indifference estimators for the slope of a univariate linear model. J Nonparametric Stat 28:395–412MathSciNetCrossRefzbMATHGoogle Scholar
  5. Cifarelli DM, Conti PL, Regazzini E (1996) On the asymptotic distribution of a general measure of monotone dependence. Ann Stat 24:1386–1399MathSciNetCrossRefzbMATHGoogle Scholar
  6. Conti PL, Nikitin YY (1999) Asymptotic efficiency of independence tests based on Gini’s rank association coefficient, Spearman’s footrule and their generalizations. Commun Stat A Theor 28:453–465CrossRefzbMATHGoogle Scholar
  7. Durante F, Fuchs S (2019) Reflection invariant copulas. Fuzzy Sets Syst 354:63–73CrossRefGoogle Scholar
  8. Durante F, Sempi C (2016) Principles of copula theory. Chapman & Hall, LondonzbMATHGoogle Scholar
  9. Edwards HH, Taylor MD (2009) Characterization of degree one bivariate measures of concordance. J Multivar Anal 100:1777–1791MathSciNetCrossRefzbMATHGoogle Scholar
  10. Edwards HH, Mikusiński P, Taylor MD (2004) Measures of concordance determined by \(D_4\)-invariant copulas. Int J Math Math Sci 70:3867–3875CrossRefzbMATHGoogle Scholar
  11. Edwards HH, Mikusiński P, Taylor MD (2005) Measures of concordance determined by \(D_4\)-invariant measures on \((0,1)^2\). Proc Am Math Soc 133:1505–1513CrossRefzbMATHGoogle Scholar
  12. Fuchs S, Schmid KD (2014) Bivariate copulas: transformations, asymmetry and measures of concordance. Kybernetika 50:109–125MathSciNetzbMATHGoogle Scholar
  13. Genest C, Verret F (2005) Locally most powerful rank tests of independence for copula models. J Nonparametric Stat 17:521–539MathSciNetCrossRefzbMATHGoogle Scholar
  14. Genest C, Quessy JF, Rémillard B (2006) Local efficiency of a Cramér-von Mises test of independence. J Multivariate Anal 97:274–294MathSciNetCrossRefzbMATHGoogle Scholar
  15. Hájek J, Šidák Z, Sen PK (1999) Theory of rank tests, 2nd edn. Academic Press, San DiegozbMATHGoogle Scholar
  16. Klement EP, Mesiar R, Pap E (2002) Invariant copulas. Kybernetika 38:275–285MathSciNetzbMATHGoogle Scholar
  17. Liebscher E (2014) Copula-based dependence measures. Depend Model 2:49–64zbMATHGoogle Scholar
  18. Nelsen RB (1998) Concordance and Gini’s measure of association. J Nonparametric Stat 9:227–238MathSciNetCrossRefzbMATHGoogle Scholar
  19. Nelsen RB (2006) An introduction to copulas. Springer, New YorkzbMATHGoogle Scholar
  20. Scarsini M (1984) On measures of concordance. Stochastica 8:201–218MathSciNetzbMATHGoogle Scholar
  21. Schweizer B, Wolff EF (1981) On nonparametric measures of dependence for random variables. Ann Stat 9:879–885MathSciNetCrossRefzbMATHGoogle Scholar
  22. Shirahata S, Wakimoto K (1984) Asymptotic normality of a class of nonlinear rank tests for independence. Ann Stat 12:1124–1129MathSciNetCrossRefzbMATHGoogle Scholar
  23. Taylor MD (2016) Multivariate measures of concordance for copulas and their marginals. Depend Model 4:224–236MathSciNetzbMATHGoogle Scholar
  24. Tchen AH (1980) Inequalities for distributions with given marginals. Ann Probab 8:814–827MathSciNetCrossRefzbMATHGoogle Scholar
  25. Van der Vaart A (1998) Asymptotic statistics. Cambridge University Press, New YorkCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Statistics and Quantitative MethodsUniversity of Milano-BicoccaMilanItaly

Personalised recommendations