Quasi-copulas with a given sub-diagonal section. (English) Zbl 1151.62044
Summary: As is well known, the Fréchet-Hoeffding bounds are the best possible for both copulas and quasi-copulas: for every (quasi-) copula \(Q\), \(\max\{x+y - 1,0\}\leq Q(x,y)\leq \min\{x,y\}\) for all \(x,y\in \)[0,1]. Sharper bounds hold when the (quasi-)copulas take prescribed values, e.g., along their diagonal or horizontal resp. vertical sections. 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 \(x_0 \in ]0,1[\). Then, we determine the best-possible bounds for sets of quasi-copulas with a given sub-diagonal section.
MSC:
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
| 60E15 | Inequalities; stochastic orderings |
| 60E05 | Probability distributions: general theory |
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