Statistical Science

Extremal Dependence Concepts

Giovanni Puccetti and Ruodu Wang

Full-text: Open access


The probabilistic characterization of the relationship between two or more random variables calls for a notion of dependence. Dependence modeling leads to mathematical and statistical challenges, and recent developments in extremal dependence concepts have drawn a lot of attention to probability and its applications in several disciplines. The aim of this paper is to review various concepts of extremal positive and negative dependence, including several recently established results, reconstruct their history, link them to probabilistic optimization problems, and provide a list of open questions in this area. While the concept of extremal positive dependence is agreed upon for random vectors of arbitrary dimensions, various notions of extremal negative dependence arise when more than two random variables are involved. We review existing popular concepts of extremal negative dependence given in literature and introduce a novel notion, which in a general sense includes the existing ones as particular cases. Even if much of the literature on dependence is focused on positive dependence, we show that negative dependence plays an equally important role in the solution of many optimization problems. While the most popular tool used nowadays to model dependence is that of a copula function, in this paper we use the equivalent concept of a set of rearrangements. This is not only for historical reasons. Rearrangement functions describe the relationship between random variables in a completely deterministic way, allow a deeper understanding of dependence itself, and have several advantages on the approximation of solutions in a broad class of optimization problems.

Article information

Statist. Sci., Volume 30, Number 4 (2015), 485-517.

First available in Project Euclid: 9 December 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Rearrangement copulas comonotonicity countermonotonicity pairwise countermonotonicity joint mixability $\Sigma$-countermonotonicity


Puccetti, Giovanni; Wang, Ruodu. Extremal Dependence Concepts. Statist. Sci. 30 (2015), no. 4, 485--517. doi:10.1214/15-STS525.

Export citation


  • Ahn, J. Y. (2015). Negative dependence concept in copulas and the marginal free herd behavior index. J. Comput. Appl. Math. 288 304–322.
  • Bernard, C., Jiang, X. and Wang, R. (2014). Risk aggregation with dependence uncertainty. Insurance Math. Econom. 54 93–108.
  • Bernard, C., Rüschendorf, L. and Vanduffel, S. (2015). Value-at-risk bounds with variance constraints. Journal of Risk and Insurance. To appear. Available at
  • Bignozzi, V. and Puccetti, G. (2015). Studying mixability with supermodular aggregating functions. Statist. Probab. Lett. 100 48–55.
  • Brewer, K. R. W. and Hanif, M. (1983). Sampling with Unequal Probabilities. Lecture Notes in Statistics 15. Springer, New York.
  • Cambanis, S., Simons, G. and Stout, W. (1976). Inequalities for $Ek(X,Y)$ when the marginals are fixed. Z. Wahrsch. Verw. Gebiete 36 285–294.
  • Cheung, K. C. (2010). Characterizing a comonotonic random vector by the distribution of the sum of its components. Insurance Math. Econom. 47 130–136.
  • Cheung, K. C. and Lo, A. (2014). Characterizing mutual exclusivity as the strongest negative multivariate dependence structure. Insurance Math. Econom. 55 180–190.
  • Clemen, R. and Reilly, T. (1999). Correlations and copulas for decision and risk analysis. Management Sci. 45 208–224.
  • Cuesta-Albertos, J. A., Rüschendorf, L. and Tuero-Díaz, A. (1993). Optimal coupling of multivariate distributions and stochastic processes. J. Multivariate Anal. 46 335–361.
  • Dall’Aglio, G. (1959). Sulla compatibilità delle funzioni di ripartizione doppia. Rend. Mat. e Appl. (5) 18 385–413.
  • Dall’Aglio, G. (1972). Fréchet classes and compatibility of distribution functions. In Symposia Mathematica, Vol. IX (Convegno di Calcolo delle Probabilità, INDAM, Rome, 1971) 131–150. Academic Press, London.
  • Day, P. W. (1972). Rearrangement inequalities. Canad. J. Math. 24 930–943.
  • Dhaene, J. and Denuit, M. (1999). The safest dependence structure among risks. Insurance Math. Econom. 25 11–21.
  • Dhaene, J., Denuit, M., Goovaerts, M. J., Kaas, R. and Vyncke, D. (2002). The concept of comonotonicity in actuarial science and finance: Theory. Insurance Math. Econom. 31 3–33.
  • Dhaene, J., Vanduffel, S., Goovaerts, M. J., Kaas, R., Tang, Q. and Vyncke, D. (2006). Risk measures and comonotonicity: A review. Stoch. Models 22 573–606.
  • Durante, F. and Fernández-Sánchez, J. (2012). On the approximation of copulas via shuffles of Min. Statist. Probab. Lett. 82 1761–1767.
  • Durante, F., Fernández-Sánchez, J. and Sempi, C. (2012). Sklar’s theorem obtained via regularization techniques. Nonlinear Anal. 75 769–774.
  • Durante, F. and Sempi, C. (2015). Principles of Copula Theory. CRC/Chapman & Hall, London.
  • Embrechts, P. and Hofert, M. (2013). A note on generalized inverses. Math. Methods Oper. Res. 77 423–432.
  • Embrechts, P., McNeil, A. J. and Straumann, D. (2002). Correlation and dependence in risk management: Properties and pitfalls. In Risk Management: Value at Risk and Beyond (Cambridge, 1998) 176–223. Cambridge Univ. Press, Cambridge.
  • Embrechts, P. and Puccetti, G. (2006). Bounds for functions of dependent risks. Finance Stoch. 10 341–352.
  • Embrechts, P., Puccetti, G. and Rüschendorf, L. (2013). Model uncertainty and VaR aggregation. J. Bank. Financ. 37 2750–2764.
  • Embrechts, P., Puccetti, G., Rüschendorf, L., Wang, R. and Beleraj, A. (2014). An academic response to Basel 3.5. Risks 2 25–48.
  • Fernández-Sánchez, J. and Trutschnig, W. (2015). Conditioning-based metrics on the space of multivariate copulas and their interrelation with uniform and levelwise convergence and iterated function systems. J. Theoret. Probab. 28 1311–1336.
  • Frank, M. J., Nelsen, R. B. and Schweizer, B. (1987). Best-possible bounds for the distribution of a sum—a problem of Kolmogorov. Probab. Theory Related Fields 74 199–211.
  • Fréchet, M. (1951). Sur les tableaux de corrélation dont les marges sont données. Ann. Univ. Lyon Sect. A (3) 14 53–77.
  • Gaffke, N. and Rüschendorf, L. (1981). On a class of extremal problems in statistics. Math. Operationsforsch. Statist. Ser. Optim. 12 123–135.
  • Gangbo, W. and McCann, R. J. (1996). The geometry of optimal transportation. Acta Math. 177 113–161.
  • Gudendorf, G. and Segers, J. (2010). Extreme-value copulas. In Copula Theory and Its Applications. Lect. Notes Stat. Proc. 198 127–145. Springer, Heidelberg.
  • Hammersley, J. M. and Morton, K. W. (1956). A new Monte Carlo technique: Antithetic variates. Proc. Cambridge Philos. Soc. 52 449–475.
  • Hanif, M. and Brewer, K. R. W. (1980). Sampling with unequal probabilities without replacement: A review. Int. Stat. Rev. 48 317–335.
  • Hardy, G. H., Littlewood, J. E. and Pólya, G. (1934). Inequalities. Cambridge Univ. Press, Cambridge.
  • Haus, U.-U. (2015). Bounding stochastic dependence, joint mixability of matrices, and multidimensional bottleneck assignment problems. Oper. Res. Lett. 43 74–79.
  • Heilmann, W.-R. (1986). On the impact of independence of risks on stop loss premiums. Insurance Math. Econom. 5 197–199.
  • Hoeffding, W. (1940). Maszstabinvariante Korrelationstheorie. Schr. Math. Inst. U. Inst. Angew. Math. Univ. Berlin 5 181–233.
  • Hoeffding, W. (1994). Scale–invariant correlation theory. In The Collected Works of Wassily Hoeffding (N. I. Fisher and P. K. Sen, eds.) 57–107. Springer, New York.
  • Hu, T. and Wu, Z. (1999). On dependence of risks and stop-loss premiums. Insurance Math. Econom. 24 323–332.
  • Jaworski, P., Durante, F., Härdle, W. and Rychlik, T., eds. (2010). Copula theory and its applications. In Proceedings of the Workshop Held at the University of Warsaw, Warsaw, September 2526, 2009. Lecture Notes in Statistics 198. Springer, Heidelberg.
  • Joe, H. (1990). Multivariate concordance. J. Multivariate Anal. 35 12–30.
  • Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London.
  • Joe, H. (2015). Dependence Modeling with Copulas. CRC Press, Boca Raton, FL.
  • Kaas, R., Dhaene, J., Vyncke, D., Goovaerts, M. J. and Denuit, M. (2002). A simple geometric proof that comonotonic risks have the convex-largest sum. Astin Bull. 32 71–80.
  • Kellerer, H. G. (1984). Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete 67 399–432.
  • Kimeldorf, G. and Sampson, A. R. (1978). Monotone dependence. Ann. Statist. 6 895–903.
  • Knott, M. and Smith, C. (2006). Choosing joint distributions so that the variance of the sum is small. J. Multivariate Anal. 97 1757–1765.
  • Kolesárová, A., Mesiar, R. and Sempi, C. (2008). Measure-preserving transformations, copulæ and compatibility. Mediterr. J. Math. 5 325–339.
  • Kolesárová, A., Mesiar, R., Mordelová, J. and Sempi, C. (2006). Discrete copulas. IEEE Trans. Fuzzy Syst. 14 698–705.
  • Kusuoka, S. (2001). On law invariant coherent risk measures. In Advances in Mathematical Economics, Vol. 3. Adv. Math. Econ. 3 83–95. Springer, Tokyo.
  • Lai, T. L. and Robbins, H. (1978). A class of dependent random variables and their maxima. Z. Wahrsch. Verw. Gebiete 42 89–111.
  • Lancaster, H. O. (1963). Correlation and complete dependence of random variables. Ann. Math. Statist. 34 1315–1321.
  • Lee, W. and Ahn, J. Y. (2014). On the multidimensional extension of countermonotonicity and its applications. Insurance Math. Econom. 56 68–79.
  • Lindvall, T. (1992). Lectures on the Coupling Method. Wiley, New York.
  • London, D. (1970). Rearrangement inequalities involving convex functions. Pacific J. Math. 34 749–753.
  • Lorentz, G. G. (1953). An inequality for rearrangements. Amer. Math. Monthly 60 176–179.
  • Mai, J.-F. and Scherer, M. (2013). What makes dependence modeling challenging? Pitfalls and ways to circumvent them. Stat. Risk Model. 30 287–306.
  • Mai, J.-F. and Scherer, M. (2014). Financial Engineering with Copulas Explained. Palgrave Macmillan, New York.
  • Makarov, G. D. (1981). Estimates for the distribution function of the sum of two random variables with given marginal distributions. Theory Probab. Appl. 26 803–806.
  • Marshall, A. W. and Olkin, I. (1983). Domains of attraction of multivariate extreme value distributions. Ann. Probab. 11 168–177.
  • Marshall, A. W., Olkin, I. and Arnold, B. C. (2011). Inequalities: Theory of Majorization and Its Applications, 2nd ed. Springer, New York.
  • McCann, R. J. (1995). Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80 309–323.
  • McNeil, A. J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton Univ. Press, Princeton, NJ.
  • Meilijson, I. and Nádas, A. (1979). Convex majorization with an application to the length of critical paths. J. Appl. Probab. 16 671–677.
  • Mikosch, T. (2006). Copulas: Tales and facts. Extremes 9 55–62.
  • Mikusiński, P., Sherwood, H. and Taylor, M. D. (1992). Shuffles of Min. Stochastica 13 61–74.
  • Minty, G. J. (1962). Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29 341–346.
  • Minty, G. J. (1964). On the monotonicity of the gradient of a convex function. Pacific J. Math. 14 243–247.
  • Müller, A. and Scarsini, M. (2000). Some remarks on the supermodular order. J. Multivariate Anal. 73 107–119.
  • Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. Wiley, Chichester.
  • Nelsen, R. B. (2006). An Introduction to Copulas, 2nd ed. Springer, New York.
  • Parthasarathy, K. R. (1967). Probability Measures on Metric Spaces. Academic Press, New York.
  • Pass, B. (2015). Multi-marginal optimal transport: Theory and applications. ESAIM Math. Model. Numer. Anal. To appear. DOI:10.1051/m2an/2015020.
  • Puccetti, G. and Rüschendorf, L. (2012). Computation of sharp bounds on the distribution of a function of dependent risks. J. Comput. Appl. Math. 236 1833–1840.
  • Puccetti, G. and Rüschendorf, L. (2013). Sharp bounds for sums of dependent risks. J. Appl. Probab. 50 42–53.
  • Puccetti, G. and Scarsini, M. (2010). Multivariate comonotonicity. J. Multivariate Anal. 101 291–304.
  • Puccetti, G. and Wang, R. (2015). Detecting complete and joint mixability. J. Comput. Appl. Math. 280 174–187.
  • Puccetti, G., Wang, B. and Wang, R. (2012). Advances in complete mixability. J. Appl. Probab. 49 430–440.
  • Puccetti, G., Wang, B. and Wang, R. (2013). Complete mixability and asymptotic equivalence of worst-possible VaR and ES estimates. Insurance Math. Econom. 53 821–828.
  • Rachev, S. T. and Rüschendorf, L. (1998). Mass Transportation Problems. Vol. I. Springer, New York.
  • Rohlin, V. A. (1952). On the fundamental ideas of measure theory. Amer. Math. Soc. Translation 1952 1–55.
  • Ruderman, H. D. (1952). Two new inequalities. Amer. Math. Monthly 59 29–32.
  • Rüschendorf, L. (1980). Inequalities for the expectation of $\Delta$-monotone functions. Z. Wahrsch. Verw. Gebiete 54 341–349.
  • Rüschendorf, L. (1981). Sharpness of Fréchet-bounds. Z. Wahrsch. Verw. Gebiete 57 293–302.
  • Rüschendorf, L. (1982). Random variables with maximum sums. Adv. in Appl. Probab. 14 623–632.
  • Rüschendorf, L. (1983). Solution of a statistical optimization problem by rearrangement methods. Metrika 30 55–61.
  • Rüschendorf, L. (2004). Comparison of multivariate risks and positive dependence. J. Appl. Probab. 41 391–406.
  • Rüschendorf, L. (2009). On the distributional transform, Sklar’s theorem, and the empirical copula process. J. Statist. Plann. Inference 139 3921–3927.
  • Rüschendorf, L. (2013). Mathematical Risk Analysis: Dependence, Risk Bounds, Optimal Allocations and Portfolios. Springer, Heidelberg.
  • Rüschendorf, L. and Uckelmann, L. (2002). On the $n$-coupling problem. J. Multivariate Anal. 81 242–258.
  • Rychlik, T. (1996). Order statistics of variables with given marginal distributions. In Distributions with Fixed Marginals and Related Topics (Seattle, WA, 1993) (L. Rüschendorf, B. Schweizer and M. Taylor, eds.). IMS Lecture Notes—Monograph Series 28 297–306. IMS, Hayward, CA.
  • Schmeidler, D. (1986). Integral representation without additivity. Proc. Amer. Math. Soc. 97 255–261.
  • Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica 57 571–587.
  • Schmid, F. and Schmidt, R. (2007). Multivariate extensions of Spearman’s rho and related statistics. Statist. Probab. Lett. 77 407–416.
  • Schweizer, B. and Sklar, A. (1974). Operations on distribution functions not derivable from operations on random variables. Studia Math. 52 43–52.
  • Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.
  • Sklar, M. (1959). Fonctions de répartition à $n$ dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 229–231.
  • Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36 423–439.
  • Tchen, A. H. (1980). Inequalities for distributions with given marginals. Ann. Probab. 8 814–827.
  • Trutschnig, W. and Fernández-Sánchez, J. (2013). Some results on shuffles of two-dimensional copulas. J. Statist. Plann. Inference 143 251–260.
  • Vijavan, K. (1968). An exact $\pi ps$ sampling scheme-generalization of a method of Hanurav. J. R. Soc. Ser. B. 30 556–566.
  • Vitale, R. A. (1979). Regression with given marginals. Ann. Statist. 7 653–658.
  • Vorob’ev, N. N. (1962). Consistent families of measures and their extensions. Theory Probab. Appl. 7 147–163.
  • Wang, R. (2014). Asymptotic bounds for the distribution of the sum of dependent random variables. J. Appl. Probab. 51 780–798.
  • Wang, R., Peng, L. and Yang, J. (2013). Bounds for the sum of dependent risks and worst value-at-risk with monotone marginal densities. Finance Stoch. 17 395–417.
  • Wang, B. and Wang, R. (2011). The complete mixability and convex minimization problems with monotone marginal densities. J. Multivariate Anal. 102 1344–1360.
  • Wang, B. and Wang, R. (2015a). Joint mixability. Mathematics of Operations Research. Available at
  • Wang, B. and Wang, R. (2015b). Extreme negative dependence and risk aggregation. J. Multivariate Anal. 136 12–25.
  • Whitt, W. (1976). Bivariate distributions with given marginals. Ann. Statist. 4 1280–1289.
  • Williamson, R. C. and Downs, T. (1990). Probabilistic arithmetic. I. Numerical methods for calculating convolutions and dependency bounds. Internat. J. Approx. Reason. 4 89–158.
  • Yaari, M. E. (1987). The dual theory of choice under risk. Econometrica 55 95–115.
  • Zarantonello, E. (1960). Solving functional equations by contractive averaging. MRC technical summary Report no. 160, University of Wisconsin, Mathematics Research Center, United States Army.