The (extended) composition of two copulas (Salvadori et al., 2006, p. 266, prop. C.4) provides for even more sophisticated structures of dependence between variables than two-copula composition in composite2COP. Let \(\mathbf{A}\) and \(\mathbf{B}\) be copulas with respective parameters \(\Theta_\mathbf{A}\) and \(\Theta_\mathbf{B}\), then
$$\mathbf{C}_{\alpha,\beta,\kappa,\gamma}(u,v) = u^\kappa v^\gamma \cdot \mathbf{A}([u^{1-\kappa}]^\alpha, [v^{1-\gamma}]^\beta) \cdot \mathbf{B}([u^{1-\kappa}]^{1-\alpha},[v^{1-\gamma}]^{1-\beta})\mbox{,}$$
defines a family of copulas \(\mathbf{C}_{\alpha,\beta,\kappa,\gamma}\) with four compositing parameters \(\alpha,\beta,\kappa,\gamma \in (0,1)\).
It is important to stress that copulas \(\mathbf{A}_{\Theta_A}\) and \(\mathbf{B}_{\Theta_B}\) can be of different families and each parameterized accordingly by the vectors of parameters \(\Theta_A\) and \(\Theta_B\).
composite3COP(u, v, para, ...)A value for the composited copula is returned.
Nonexceedance probability \(u\) in \(X\) direction;
Nonexceedance probability \(v\) in \(Y\) direction;
A special parameter list (see Note); and
Additional arguments to pass to composite2COP.
W.H. Asquith
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in Nature---An approach using copulas: Springer, 289 p.
COP, breveCOP, simCOP, composite1COP, composite2COP, convexCOP, glueCOP, simcomposite3COP