Mathematical background

Copulas and Sklar's theorem

Let   be a random vector with joint distribution function and marginal distribution functions . Sklar's theorem states that there exists a copula such that

If the margins are continuous, the copula is unique. This separates marginal modeling from dependence modeling. In practice, one often transforms data to pseudo-observations

or obtains   from fitted marginal models. VineCopulas.jl works on the copula scale, so input data should already be in .

Copulas.jl provides the general copula layer, pseudo-observation utilities and SklarDist; VineCopulas.jl focuses on the vine composition layer.

Why vines?

A direct -dimensional parametric copula may be too rigid. Vine copulas address this by decomposing the joint copula density into bivariate building blocks. These bivariate blocks can come from different families and can capture different dependence patterns across pairs and conditional pairs.

Under the simplifying assumption, the conditional pair-copulas do not vary with the actual value of the conditioning variables. This is the standard simplified vine copula model implemented by VineCopulas.jl.

The copula scale

A vine copula density is a density on the unit hypercube. It is not a full model for raw data until combined with marginal densities:

This package implements and its simulation/transformation methods. Marginal modeling belongs upstream.

Dependence summaries

Common dependence summaries include Kendall's , Spearman's , and tail-dependence coefficients. They are useful for exploratory analysis and structure selection, but VineCopulas.jl v0.1 does not yet implement automatic selection based on them. For now, the user constructs the vine explicitly.