On Minimal Copulas under the Concordance Order

Title
On Minimal Copulas under the Concordance Order
Authors
Ahn, Jae YounFuchs, Sebastian
Ewha Authors
SCOPUS Author ID
안재윤
Issue Date
2020
Journal Title
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
ISSN
0022-3239

1573-2878
Citation
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS vol. 184, no. 3, pp. 762 - 780
Keywords
Concordance orderCountermonotonicityExtreme negative dependenceKendall's tauMinimal copulaOptimizationSpearman's rho
Publisher
SPRINGER/PLENUM PUBLISHERS
Indexed
SCI; SCIE; SCOPUS
Document Type
Article
Abstract
In the present paper, we study extreme negative dependence focussing on the concordance order for copulas. With the absence of a least element for dimensions d >= 3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d\ge 3$$\end{document}, the set of all minimal elements in the collection of all copulas turns out to be a natural and quite important extreme negative dependence concept. We investigate several sufficient conditions, and we provide a necessary condition for a copula to be minimal. The sufficient conditions are related to the extreme negative dependence concept of d-countermonotonicity and the necessary condition is related to the collection of all copulas minimizing multivariate Kendall's tau. The concept of minimal copulas has already been proved to be useful in various continuous and concordance order preserving optimization problems including variance minimization and the detection of lower bounds for certain measures of concordance. We substantiate this key role of minimal copulas by showing that every continuous and concordance order preserving functional on copulas is minimized by some minimal copula, and, in the case the continuous functional is even strictly concordance order preserving, it is minimized by minimal copulas only. Applying the above results, we may conclude that every minimizer of Spearman's rho is also a minimizer of Kendall's tau.
DOI
10.1007/s10957-019-01618-4
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자연과학대학 > 통계학전공 > Journal papers
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