Abstract
We determine two constructions that, starting with two bivariate copulas, give rise to new bivariate and trivariate copulas, respectively. These constructions are used to determine pointwise upper and lower bounds for the class of all trivariate copulas with given bivariate marginals.
Highlights
Several researchers have focused the attention on constructions and stochastic orders among probability distribution functions with given marginals
Instead, we have some information about the multivariate marginals of F, the problem has not been considered extensively in the literature, it seems natural that for some applications one needs to estimate the joint distribution F of X, when the dependence among some components of F is known
We refer to Ruschendorf 2, 3 and Joe 4, 5
Summary
Several researchers have focused the attention on constructions and stochastic orders among probability distribution functions with given marginals. We aim at contributing to this problem by providing lower and upper bounds in the class of continuous trivariate d.f.’s whose bivariate marginals are given, that is, when we have full information about the pairwise dependence among the components of the corresponding random vector. These new bounds improve some estimations given by Joe. We will formulate our results in the class of copulas, which are multivariate d.f.’s whose one-dimensional marginals are uniformly distributed on 0, 1 : see Joe 5 ; Nelsen 6. These constructions can be seen as generalizations of the product-like operations on copulas considered by Darsow et al 8 and Kolesarovaet al. 9
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