In a 2-dimensional space, Fréchet-Hoeffding upper and lower bounds define comonotonicity and countermonotonicity, respectively. Similarly, in the multidimensional case, comonotonicity can be defined using the Fréchet-Hoeffding upper bound. However, since the multidimensional Fréchet-Hoeffding lower bound is not a distribution function, there is no obvious extension of countermonotonicity in multidimensions. This paper investigates in depth a new multidimensional extension of countermonotonicity. We first provide an equivalent condition for countermonotonicity in 2-dimension, and extend the definition of countermonotonicity into multidimensions. In order to justify such extensions, we show that newly defined countermonotonic copulas constitute a minimal class of copulas. Two applications will be provided. First, we will study the relationships between multidimensional countermonotonicity and such well-known multivariate concordance measures as Kendall's tau or Spearman's rho. Second, we will give a financial interpretation of multidimensional countermonotonicity via the existing herd behavior index.
Bibliographical noteFunding Information:
For Woojoo Lee, this work was supported by Inha University Research Grant. For Jae Youn Ahn, this work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government ( 2013R1A1A1076062 ) and Ewha Womans University Research Grant of 2013.
- Herd behavior index
- Measures of concordance
- Minimal copula