The Annals of Applied Probability

Bernoulli and tail-dependence compatibility

Paul Embrechts, Marius Hofert, and Ruodu Wang

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The tail-dependence compatibility problem is introduced. It raises the question whether a given $d\times d$-matrix of entries in the unit interval is the matrix of pairwise tail-dependence coefficients of a $d$-dimensional random vector. The problem is studied together with Bernoulli-compatible matrices, that is, matrices which are expectations of outer products of random vectors with Bernoulli margins. We show that a square matrix with diagonal entries being 1 is a tail-dependence matrix if and only if it is a Bernoulli-compatible matrix multiplied by a constant. We introduce new copula models to construct tail-dependence matrices, including commonly used matrices in statistics.

Article information

Ann. Appl. Probab. Volume 26, Number 3 (2016), 1636-1658.

Received: January 2015
First available in Project Euclid: 14 June 2016

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Zentralblatt MATH identifier

Primary: 60E05: Distributions: general theory 62H99: None of the above, but in this section 62H20: Measures of association (correlation, canonical correlation, etc.) 62E15: Exact distribution theory 62H86: Multivariate analysis and fuzziness

Tail dependence Bernoulli random vectors compatibility matrices copulas insurance application


Embrechts, Paul; Hofert, Marius; Wang, Ruodu. Bernoulli and tail-dependence compatibility. Ann. Appl. Probab. 26 (2016), no. 3, 1636--1658. doi:10.1214/15-AAP1128.

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  • [1] Berman, A. and Shaked-Monderer, N. (2003). Completely Positive Matrices. World Scientific, River Edge, NJ.
  • [2] Bluhm, C. and Overbeck, L. (2007). Structured Credit Portfolio Analysis, Baskets & CDOs. Chapman & Hall/CRC, Boca Raton, FL.
  • [3] Bluhm, C., Overbeck, L. and Wagner, C. (2002). An Introduction to Credit Risk Modeling. Chapman & Hall, London.
  • [4] Chaganty, N. R. and Joe, H. (2006). Range of correlation matrices for dependent Bernoulli random variables. Biometrika 93 197–206.
  • [5] Dhaene, J. and Denuit, M. (1999). The safest dependence structure among risks. Insurance Math. Econom. 25 11–21.
  • [6] 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.
  • [7] Fiebig, U., Strokorb, K. and Schlather, M. (2014). The realization problem for tail correlation functions. Available at
  • [8] Joe, H. (1997). Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability 73. Chapman & Hall, London.
  • [9] Joe, H. (2015). Dependence Modeling with Copulas. Monographs on Statistics and Applied Probability 134. CRC Press, Boca Raton, FL.
  • [10] Kortschak, D. and Albrecher, H. (2009). Asymptotic results for the sum of dependent non-identically distributed random variables. Methodol. Comput. Appl. Probab. 11 279–306.
  • [11] Liebscher, E. (2008). Construction of asymmetric multivariate copulas. J. Multivariate Anal. 99 2234–2250.
  • [12] McNeil, A. J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton Univ. Press, Princeton, NJ.
  • [13] Nikoloulopoulos, A. K., Joe, H. and Li, H. (2009). Extreme value properties of multivariate $t$ copulas. Extremes 12 129–148.
  • [14] Rüschendorf, L. (1981). Characterization of dependence concepts in normal distributions. Ann. Inst. Statist. Math. 33 347–359.
  • [15] Strokorb, K., Ballani, F. and Schlather, M. (2015). Tail correlation functions of max-stable processes: Construction principles, recovery and diversity of some mixing max-stable processes with identical TCF. Extremes 18 241–271.
  • [16] Yang, J., Qi, Y. and Wang, R. (2009). A class of multivariate copulas with bivariate Fréchet marginal copulas. Insurance Math. Econom. 45 139–147.