An overview of the goodness-of-fit test problem for copulas

We review the main "omnibus procedures" for goodness-of-fit testing for copulas: tests based on the empirical copula process, on probability integral transformations, on Kendall's dependence function, etc, and some corresponding reductions of dimension techniques. The problems of finding asymptotic distribution-free test statistics and the calculation of reliable p-values are discussed. Some particular cases, like convenient tests for time-dependent copulas, for Archimedean or extreme-value copulas, etc, are dealt with. Finally, the practical performances of the proposed approaches are briefly summarized

Final solution to the problem of relating a true copula to an imprecise copula

In this paper we solve in the negative the problem proposed in this journal (I. Montes et al., Sklar's theorem in an imprecise setting, Fuzzy Sets and Systems, 278 (2015), 48-66) whether an order interval defined by an imprecise copula contains a copula. Namely, if $\mathcal{C}$ is a nonempty set of copulas, then $\underline{C} = \inf\{C\}_{C\in\mathcal{C}}$ and $\overline{C}= \sup\{C\}_{C\in\mathcal{C}}$ are quasi-copulas and the pair $(\underline{C},\overline{C})$ is an imprecise copula according to the definition introduced in the cited paper, following the ideas of $p$-boxes. We show that there is an imprecise copula $(A,B)$ in this sense such that there is no copula $C$ whatsoever satisfying $A \leqslant C\leqslant B$. So, it is questionable whether the proposed definition of the imprecise copula is in accordance with the intentions of the initiators. Our methods may be of independent interest: We upgrade the ideas of Dibala et al. (Defects and transformations of quasi-copulas, Kybernetika, 52 (2016), 848-865) where possibly negative volumes of quasi-copulas as defects from being copulas were studied.Comment: 20 pages; added Conclusion, added some clarifications in proofs, added some explanations at the beginning of each section, corrected typos, results remain the sam

Dependent jump processes with coupled Lévy measures

I present a simple method for the modeling and simulation of dependent positive jump processes through a series representation. Each constituent process is represented by a series whose terms are equal to a transformation of the jump times of a standard Poisson process. The transformations are given by the inverses of the respective marginal Lévy tail mass integral functions. The dependence between the various constituent processes is given by a probabilistic copula for the inter-arrival times of the various standard Poisson processes.Lévy copulas, Copulas, Lévy processes, Monte-Carlo simulations

Distorted Copulas: Constructions and Tail Dependence

Given a copula C, we examine under which conditions on an order isomorphism ψ of [0, 1] the distortion C ψ: [0, 1]2 → [0, 1], C ψ(x, y) = ψ{C[ψ−1(x), ψ−1(y)]} is again a copula. In particular, when the copula C is totally positive of order 2, we give a sufficient condition on ψ that ensures that any distortion of C by means of ψ is again a copula. The presented results allow us to introduce in a more flexible way families of copulas exhibiting different behavior in the tails