Quasi-copulas with a given sub-diagonal section

As is well known, the Fréchet–Hoeffding bounds are the best possible for both copulas and quasi-copulas: for every (quasi-) copula Q , max { x + y − 1 , 0 } ≤ Q ( x , y ) ≤ min { x , y } for all x , y ∈ [ 0 , 1 ] . Sharper bounds hold when the (quasi-)copulas take prescribed values, e.g., along thei...

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Bibliographic Details
Published in:Nonlinear analysis 2008-12, Vol.69 (12), p.4654-4673
Main Authors: Quesada-Molina, José Juan, Saminger-Platz, Susanne, Sempi, Carlo
Format: Article
Language:English
Subjects:
Copula
Exact sciences and technology
Mathematical analysis
Mathematics
Quasi-copula
Sciences and techniques of general use
Sub-diagonal (section)
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Summary:As is well known, the Fréchet–Hoeffding bounds are the best possible for both copulas and quasi-copulas: for every (quasi-) copula Q , max { x + y − 1 , 0 } ≤ Q ( x , y ) ≤ min { x , y } for all x , y ∈ [ 0 , 1 ] . Sharper bounds hold when the (quasi-)copulas take prescribed values, e.g., along their diagonal or horizontal resp. vertical sections. Here we pursue two goals: first, we investigate construction methods for (quasi-)copulas with a given sub-diagonal section, i.e., with prescribed values along the straight line segment joining the points ( x 0 , 0 ) and ( 1 , 1 − x 0 ) for x 0 ∈ ] 0 , 1 [ . Then, we determine the best-possible bounds for sets of quasi-copulas with a given sub-diagonal section.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2007.11.021