Multivariate countermonotonicity and the minimal copulas

Woojoo Lee; Ka Chun Cheung; Jae Youn Ahn

doi:10.1016/j.cam.2016.12.032

Multivariate countermonotonicity and the minimal copulas

Woojoo Lee, Ka Chun Cheung, Jae Youn Ahn

Department of Statistics

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	589-602
Number of pages	14
Journal	Journal of Computational and Applied Mathematics
Volume	317
DOIs	https://doi.org/10.1016/j.cam.2016.12.032
State	Published - 1 Jun 2017

Bibliographical note

Funding 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).

Publisher Copyright:
© 2017 Elsevier B.V.

Keywords

Comonotonicity
Countermonotonicity
Minimal copula
Variance minimization

Access to Document

10.1016/j.cam.2016.12.032

Cite this

@article{efae38eacf7e4fa88e0be1cf830e7ec1,

title = "Multivariate countermonotonicity and the minimal copulas",

abstract = "Fr{\'e}chet–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 Fr{\'e}chet–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.",

keywords = "Comonotonicity, Countermonotonicity, Minimal copula, Variance minimization",

author = "Woojoo Lee and Cheung, {Ka Chun} and Ahn, {Jae Youn}",

note = "Funding 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). Publisher Copyright: {\textcopyright} 2017 Elsevier B.V.",

year = "2017",

month = jun,

day = "1",

doi = "10.1016/j.cam.2016.12.032",

language = "English",

volume = "317",

pages = "589--602",

journal = "Journal of Computational and Applied Mathematics",

issn = "0377-0427",

publisher = "Elsevier",

}

TY - JOUR

T1 - Multivariate countermonotonicity and the minimal copulas

AU - Lee, Woojoo

AU - Cheung, Ka Chun

AU - Ahn, Jae Youn

N1 - Funding 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). Publisher Copyright: © 2017 Elsevier B.V.

PY - 2017/6/1

Y1 - 2017/6/1

N2 - 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.

AB - 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.

KW - Comonotonicity

KW - Countermonotonicity

KW - Minimal copula

KW - Variance minimization

UR - http://www.scopus.com/inward/record.url?scp=85008913767&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2016.12.032

DO - 10.1016/j.cam.2016.12.032

M3 - Article

AN - SCOPUS:85008913767

SN - 0377-0427

VL - 317

SP - 589

EP - 602

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

ER -

Multivariate countermonotonicity and the minimal copulas

Abstract

Bibliographical note

Keywords

Access to Document

Fingerprint

Cite this