The composition of two copulas (Salvadori et al., 2007, p. 266, prop. C.3) provides for more sophisticated structures of dependence between variables than many single parameter copula can provide. Further, asymmetrical copulas are readily obtained from symmetrical copulas. Let \(\mathbf{A}\) and \(\mathbf{B}\) be copulas with respective parameters \(\Theta_\mathbf{A}\) and \(\Theta_\mathbf{B}\), then
$$\mathbf{C}_{\alpha,\beta}(u,v) = \mathbf{A}(u^\alpha, v^\beta) \cdot \mathbf{B}(u^{1-\alpha},v^{1-\beta})\mbox{,}$$
defines a family of copulas \(\mathbf{C}_{\alpha,\beta; \Theta_\mathbf{A}, \Theta_\mathbf{B}}\) with two compositing parameters \(\alpha,\beta \in \mathcal{I}:[0,1]\). In particular if \(\alpha = \beta = 1\), then \(\mathbf{C}_{1,1} = \mathbf{A}\), and if \(\alpha = \beta = 0\), then \(\mathbf{C}_{0,0} = \mathbf{B}\). For \(\alpha \ne \beta\), the \(\mathbf{C}_{\alpha,\beta}\) is in general asymmetric that is \(\mathbf{C}(u,v) \ne \mathbf{C}(v,u)\) for some \((u,v) \in \mathcal{I}^2\). This construction technique is named Khoudraji device within the copula package (see khoudrajiCopula therein).
It is important to stress that copulas \(\mathbf{A}_{\Theta_A}\) and \(\mathbf{B}_{\Theta_B}\) can be of different families and each copula parameterized accordingly by the vector of parameters \(\Theta_A\) and \(\Theta_B\). This is an interesting feature in the context of building complex structures when pursuing asymmetric measures of dependency such as the L-comoments. Do the copulas \(\mathbf{A}\) and \(\mathbf{B}\) need be symmetric? The Salvadori reference makes no stated restriction to that effect. Symmetry of the copula \(\mathbf{C}\) is required for the situation that follows, however.
It is possible to simplify the construction of an asymmetric copula from a symmetric copula by the following. Let \(\mathbf{C}(u,v)\) be a symmetric copula, \(\mathbf{C} \ne \mathbf{\Pi}\) (for \(\mathbf{\Pi}\) see P). A family of asymmetric copulas \(\mathbf{C}_{\alpha,\beta}\) with two composition parameters \(0 < \alpha,\beta < 1, \mbox{and\ } \alpha \ne \beta\) that also includes \(\mathbf{C}(u,v)\) as a limiting case and is given by
$$\mathbf{C}_{\alpha,\beta}(u,v) = u^\alpha v^\beta \cdot \mathbf{C}(u^{1-\alpha},v^{1-\beta})\mbox{.}$$
The composite2COP function is based on the more general result given in the former rather than the later mathematical definition to provide additional flexibility. For simpler case of composition involving only one copula, composite1COP is available, and a more complex (extended) composition is available in composite3COP.
composite2COP(u, v, para, ...)Value(s) for the composited copula is returned.
Nonexceedance probability \(u\) in the \(X\) direction;
Nonexceedance probability \(v\) in the \(Y\) direction;
A special parameter list (see Note); and
Additional arguments to pass to the copulas.
W.H. Asquith
Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in Nature---An approach using copulas: Springer, 289 p.
COP, breveCOP, composite1COP, composite3COP,
convexCOP, glueCOP