Fréchet–Hoeffding upper and lower bounds play an important role in various bivariate optimization problems because they are the maximum and minimum of bivariate copulas in concordance order, respectively. However, while the Fréchet–Hoeffding upper bound is the maximum of any multivariate copulas, there is no minimum copula available for dimensions d≥3. Therefore, multivariate minimization problems with respect to a copula are not straightforward as the corresponding maximization problems. When the minimum copula is absent, minimal copulas are useful for multivariate minimization problems. We illustrate the motivation of generalizing the joint mixability to d-countermonotonicity defined in Lee and Ahn (2014) through variance minimization problems and show that d-countermonotonic copulas are minimal copulas.
Bibliographical noteFunding Information:
Woojoo Lee's work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03936100) and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2011-0030810). Ka Chun Cheung's work was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU701213P), and the CAE 2013 research grant from the Society of Actuaries. Any opinions, finding, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the SOA. Jae Youn Ahn's work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (NRF-2013R1A1A1076062).
© 2017 Elsevier B.V.
- Minimal copula
- Variance minimization