Extremes of the mass distribution associated with a trivariate quasi-copula
type: Item
instance of: scholarly article
title: Extremes of the mass distribution associated with a trivariate quasi-copula
zbMATH Open document ID: 1131.62044
author: B. De Baets
author: H. De Meyer
author: Manuel Úbeda-Flores
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2007-06-04T00:00:00Z
review text: A 3-quasicopula is a function \(Q:[0,1]^{3} \rightarrow [0,1]\), with the properties \[ Q(0,a,b)=Q(a,0,b)=Q(a,b,0)=0,\quad Q(a,1,1)=Q(1,a,1)=Q(1,1,a)=a, \] \[ \text{and}\qquad | Q((a_{i}))-Q((b_{i}))| \leq \sum | a_{i}-b_{i}|. \] A box is \[ B=\Pi_{i=1}^{3}[a_{i}^{1},a_{i}^{2}]\quad \text{with}\quad 0 \leq a_{i}^{1} < a_{i}^{2} \leq 1, \] and \(V_{Q}(B)\) is defined as \[ \sum_{i_{j}=1,2} (-1)^{\text{card} \{j;i_j=1\}}Q(a_{1}^{i_1},a_{2}^{i_2},a_{3}^{i_3}). \] First result: \(V_{Q}(B)\in [-4/5,1]\), \(V_{Q}(B)=-4/5\) implies \(B=[2/5,4/5]^{3}\), \(V_{Q}(B)=1\) implies \(B=[a,1]^{3}\), \(a\in [0,1/2]\). Examples of 3- quasicopulas \(Q\) and of \(B\) are given in which \(V_{Q}(B)=1\) and \(-4/5\). If \(a_{1}^{1}=0\) then \(V_{Q}(B)\geq -1/2\); equality is possible for \(B= [0,1/2]\times [1/2,1]^{2}\). If \(a_{1}^{1}=a_{2}^{1}=0\) then \(V_{Q}(B)\geq 0\), and equality implies \(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}-a_{3}^{1}\leq 2\). If \(a_{1}^{2}=1\) then \(V_{Q}(B)\geq -2/3\), and equality implies \[ B= [a,1]\times [1/3,2/3]^{2},\;\;a\in [1/3,2/3]. \] If \(a_{1}^{2}=a_{2}^{2}=1\) then \(V_{Q}(B)\geq -1/2\), equality requires \[ B= [1/2,1]^{2}\times [a,a+(1/2)],\;\;a\in [0,1/2]. \] All conclusions are obtained by linear programming.
Mathematics Subject Classification ID: 62H05
Mathematics Subject Classification ID: 62G32
Mathematics Subject Classification ID: 60E05
Mathematics Subject Classification ID: 90C05
Mathematics Subject Classification ID: 90C90
zbMATH DE Number: 5161246
zbMATH Keywords: quasicopula
zbMATH Keywords: 3 dimensional
zbMATH Keywords: box
zbMATH Keywords: extreme values.
Wikidata QID: Q60258001
reviewed by: Ion Cuculescu
MaRDI profile type: Publication
full work available at URL: j.crma.2007.03.026
OpenAlex ID: W2092608072
cites work: On the characterization of a class of binary operations on distribution functions
cites work: Q4779803
cites work: Q4447165
cites work: Q5692485
cites work: An introduction to copulas.
cites work: Q5386486
DOI: 10.1016/J.CRMA.2007.03.026
description: scientific article; zbMATH DE number 5161246