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Best-possible bounds on sets of bivariate distribution functions. (English) Zbl 1057.62038
The aim of this paper is to present a method for finding bounds on arbitrary sets of joint distribution functions \(H\) of continuous random variables \(X\) and \(Y\), \(H\) having the marginal distribution functions \(F\) and \(G\), respectively. The paper uses copulas \(C\) (uniquely determined by the relation \(H(x, y)= C(F(x),G(y)))\) and quasi-copulas to narrow the classical Fréchet-Hoeffding inequality (and bounds) in order to obtain pointwise best possible bounds on non-empty sets of distribution functions. As an application, the proposed procedure is illustrated by finding bounds when the values of \(H\) are known at quartiles of \(X\) and \(Y\).

MSC:
62H05 Characterization and structure theory for multivariate probability distributions; copulas
60E15 Inequalities; stochastic orderings
62E10 Characterization and structure theory of statistical distributions
62H20 Measures of association (correlation, canonical correlation, etc.)
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