Non-exchangeable random variables, Archimax copulas and their fitting to real data

In recent years copulas turned out to be a promising tool in multivariate modelling, mostly with applications in actuarial sciences and hydrology. In short, copula is a function which allows modelling dependence structure between stochastic variables. The main advantage is that the copula approach can split the problem of constructing multivariate probability distributions into a part containing the marginal one-dimensional distribution functions and a part containing the dependence structure. These two parts can be studied and estimated separately and then rejoined to form a joint distribution function. Restricting ourselves to bivariate case, copula is a function C : [0, 1] → [0, 1] which satisfies the boundary conditions, C(t, 0) = C(0, t) = 0 and C(t, 1) = C(1, t) = t for t ∈ [0, 1] (uniform margins), and the 2-increasing property, C(u2, v2)−C(u2, v1)− C(u1, v2) + C(u1, v1) ≥ 0 for all u1 ≤ u2, v1 ≤ v2 ∈ [0, 1]. Copula is symmetric if C(u, v) = C(v, u) for all (u, v) ∈ [0, 1] and is asymmetric otherwise. By [a, b] we mean a closed interval with endpoints a and b, while ]a, b[ will denote an open interval. There are several approaches how to model exchangeable random variables. Most of them refer to Archimedean copulas [15], i. e., copulas Cφ : [0, 1] 2 → [0, 1] expressible in the form Cφ(u, v) = φ(−1) (φ(u) + φ(v)) , (1)