4 research outputs found
Constructing copulas from shock models with imprecise distributions
The omnipotence of copulas when modeling dependence given marg\-inal
distributions in a multivariate stochastic situation is assured by the Sklar's
theorem. Montes et al.\ (2015) suggest the notion of what they call an
\emph{imprecise copula} that brings some of its power in bivariate case to the
imprecise setting. When there is imprecision about the marginals, one can model
the available information by means of -boxes, that are pairs of ordered
distribution functions. By analogy they introduce pairs of bivariate functions
satisfying certain conditions. In this paper we introduce the imprecise
versions of some classes of copulas emerging from shock models that are
important in applications. The so obtained pairs of functions are not only
imprecise copulas but satisfy an even stronger condition. The fact that this
condition really is stronger is shown in Omladi\v{c} and Stopar (2019) thus
raising the importance of our results. The main technical difficulty in
developing our imprecise copulas lies in introducing an appropriate stochastic
order on these bivariate objects
Semiquadratic copulas based on horizontal and vertical interpolation
We introduce several families of semiquadratic copulas (i.e. copulas that are quadratic in any point of the unit square in at least one coordinate) of which the diagonal and/or opposite diagonal sections are given functions. These copulas are constructed by quadratic interpolation on segments connecting the diagonal, opposite diagonal and sides of the unit square; all interpolations are therefore performed horizontally or vertically. For each family we provide the necessary and sufficient conditions on the given diagonal and/or opposite diagonal functions and two auxiliary real functions to obtain a copula that has these diagonal and/or opposite diagonal functions as diagonal and/or opposite diagonal sections. Just as the product copula is a central member of all families of semilinear copulas based on horizontal and vertical interpolation, it turns out that the Farlie Gumbel Morgenstern family of copulas is included in all families of semiquadratic copulas introduced and characterized here