Negative dependence concept in copulas and the marginal free herd behavior index

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Abstract We provide a set of copulas that can be interpreted as having the negative extreme dependence. This set of copulas is interesting because it coincides with countermonotonic copula for a bivariate case, and more importantly, is shown to be minimal in concordance ordering in the sense that no copula exists which is strictly smaller than the given copula outside the proposed copula set. Admitting the absence of the minimum copula in multivariate dimensions greater than 2, the study of the set of minimal copulas can be important in the investigation of various optimization problems. To demonstrate the importance of the proposed copula set, we provide the variance minimization problem of the aggregated sum with arbitrarily given uniform marginals. As a financial/actuarial application of these copulas, we define a new herd behavior index using weighted Spearman's rho, and determine the sharp lower bound of the index using the proposed set of copulas.

Original languageEnglish
Article number10133
Pages (from-to)304-322
Number of pages19
JournalJournal of Computational and Applied Mathematics
StatePublished - 23 Apr 2015

Bibliographical note

Funding Information:
The author acknowledges the reviews by two anonymous reviewers, and also likes to thank Woojoo Lee for meaningful discussions. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government ( 2013R1A1A1076062 ).

Publisher Copyright:
© 2015 Elsevier B.V.


  • Comonotonicity
  • Herd behavior index
  • Negative extreme dependence


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